DEALING WITH
DECIMALS
Prepared for the course team by John Mason
.
* Centre
,
for
. Mathematics
ducatio ion
,
Project MATHEMATICS UPDATE Course Team
Gaynor Arrowsmith, Project Officer, Open University
Lynne Burrell, Academic Editor, Open University
Leone Burton, External Assessor, Thames Polytechnic
Joy Davis, Liaison Adviser, Open University
Peter Gates, Author, Open University
Pete Griffin, Author, Open University
Nick James, Liaison Adviser, Open University
Barbara Jaworski, Author, Open University
John Mason, Author and Project Leader, Open University
Rachel Pearce, Secretary, Open University
Acknowledgments
Project MATHEMATICS UPDATE was funded by a grant from the Department of Education and
Science. We are most grateful for comments from Ruth Eagle, Arthur Hanley, Michelle
Selinger, Eileen Billington, Gillian Hatch, John Branfield and many others who may not
have realised a t the time that they were working on parts of this pack.
The Open University, Walton Hall, Milton Keynes, MK7 6AA.
First published 1989.
Copyright O 1989 The Open University.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or
transmitted, in any form or by any means, without written permission from the publisher.
Printed in Great Britain by The Open University.
Further information on this and other Open University courses may be obtained from the Learning
Materials Services Mice, Centre for Continuing Education, The Open University, PO Box 188, Milton
Keynes MK7 6DH.
ISBN 0 335 17433 7
CONTENTS
0
INTRODUCTION
AIMS
WAYS OF WORKING
1
NUMBERS ON THE LINE
Integers; Finite decimal-names; Infinite decimal-names;
Abstraction
2
MULTIPLE NAMES
16
Whole numbers; Fractions; Irrationals; Surds for the bold;
Abstraction
3
DECIMAL ARITHMETIC
28
What is the problem?; Fractions into rationals; Real numbers;
Surd arithmetic; Abstraction
4
DIFFERENT BASES
39
Integers; Parts of a whole; Exploiting different bases;
Abstraction
5
GEOMETRICAL ARITHMETIC
48
6
THE LAST WORD
52
7
GLOSSARY
53
0 INTRODUCTION
How are numbers located on a number-line?
What exactly is a decimal number?
Decimal numbers are a common part of our culture, but there are sophisticated ideas hiding
behind the innocuous-looking decimal point.
What i s the connection between decimals and whole
numbers? Decimals and fractions?
Why do pupils seem to treat them as completely separate
ideas?
What other systems are there for naming numbers?
The purpose of this pack is to explore the basic notion of decimal-names for numbers.
Emphasis is plac.ed on symbols like 21.3 as 'the name for a number rather than as the number
itself, in order to cope with the'multiplicity of different names that are used for the same
number. In the process of exploring decimals, it is quite natural that ideas about whole
numbers and fractions are encountered as well.
For a more detailed account of many of the ideas presented here see also:
'
G. Flegg, Numbers: their history and meaning, Penguin 1983;
A. ~ a r d i n e rInfinite
,
processes: background to analysis, Springer-Verlag 1982.
AIMS
To indicate various ways in which people employ and deal
with decimals.
To provide situations which will engender confidence in
working with decimals.
To study connections between integers, fractions and
decimals.
To draw attention to the arithmetic of fractions and of
decimals.
To link place-value and base-ten to decimals, and to explore
the use of other bases both for integers and for numbers
between 0 and 1.
To suggest that infinity is never far away from any
mathematical question.
WAYS OF WORKING
This pack consists of sections of related activities, each section focussing on an important
aspect of decimals in mathematics. The first section introduces decimals as names for points
on a number-line. Subsequent sections then explore ramifications of this idea, introducing
rational and irrational numbers and geometric constructions which correspond t o
arithmetical operations on numbers. Words in italics are either for emphasis, or are
technical terms which are explained in the GLOSSARY. It is not intended that you work
systematically section by section, or even that you tackle every activity in a section. Rather,
we suggest that you find some attractive activities, explore in and around them, and discuss
what you find with colleagues.
Your most important activity will be your attempt to tell yourself a coherent (mathematical)
story after working on one or more of the suggested activities, and to see how different
activities relate to each other. You cannot deem yourself to have finished until you have
completed Section 6 THE LAST WORD.
It is important when working on an activity, not to get lost simply in the 'doing'. Every so
often there are suggestions for reflection - for pausing, drawing back, and making sense of
what has been noticed. Drawing together several ideas in the mind, and trying to hold them
there, can help to integrate them into a whole. There are also some suggestions for pausing
and considering similarities and differences between activities. I t is hoped that such pausing
will become natural and automatic. It is impossible to mark in the text every useful pausing
point.
Reflection is partly a solitary activity, as you try to encompass a number of disparate ideas a t
once in your mind. It is also a communal activity, as you try to express to others the sense you
have made, in the form of connections, similarities and differences. Your attempts to express
ideas which are only half formed can certainly help you, and you can gain from the ideas and
perspectives of others. To prepare for a productive discussion with colleagues, it is usually
useful to try to write down a few sentences that capture what you think the various activities
have been about, with any questions or uncertainties that you are aware of.
Working on an activity, it is very possible that you will get stuck. What do you do then? The
BOOKMARK: What to do when you're stuck contains some suggestions. Use it. Use any
colleagues you have access to, to discuss your 'stuckness'. It may be that you are not clear what
the activity involves. If so, then investigate for yourself, formulating your own questions. It
may be that you are not clear what the activity is trying to get at. If that is the case, then try to
make up some sort of story, write it down, and compare notes with colleagues.
After working for a while on a mathematical activity, there is often a strong desire to know
whether you are 'on the right track'. However, this pack contain no answers as such, for many
reasons. The most salient reason is that generally there is no particular correct answer.
There are many things worth noticing, and these are likely to emerge from reflection and
discussion. It is essential that you develop your own criteria for deciding when you have 'done
enough for now'. If you have some nagging doubt that there is more to discover, then perhaps it
is worth going back and investigating further.
More detailed suggestions on ways of working are provided in the TUTOR PACK. In addition, the TUTOR
PACK contains notes on how to run a series of meetings based on any of the materials in the MATHEMATICS
UPDATE series. PM751 EXPRESSING GENERALITY, which is intended as a foundation to the UPDATE series,
also contains brief notes for tutors and course organisers.
1 NUMBERS ON THE LINE
This section introduces decimals as names for points on the number-line. Ramifications are
developed in later sections. I t is important to spend a t least a little time thinking about each
activity, and trying one or two in each section, because aspects of decimal-names are
introduced bit by bit. It might be sensible to pause before the subsection on infinite names and
to do some work on Fractions (in Section 2) in parallel.
Most activities in this section ask you to imagine a number-line with certain points marked
on it. It is much easier to close your eyes, and have someone read the instructions out, with
pauses a t the end of each sentence to enable you to stabilise your image. Every so often i t is
worth trying to describe to a colleague (or to the whole group) what it is that you are seeing,
trying to be brief but vivid, so that they can see what you are seeing. If a t any time you are not
sure what someone has instructed you to do, or what someone else has described, then ask them
to concentrate on describing what they are seeing. It may help to draw a number-line in order
to stabilise your mental image.
INTEGERS
An integer (from the Latin for number) is a number - named by symbols such as 2,5, 0 , 3 , etc.
Sometimes these numbers are also called whole numbers (see GLOSSARY),though some people
restrict the term 'whole number' to positive counting numbers. Integers form the starting
point for this section.
WHERE ARE THE INTEGERS? Imagine a number-line. Imagine a point marked or labelled 0
somewhere within your vision. Imagine that all the integers have been marked or labelled: 1,
2, . . ., and -1, -2, . . .. Now focus on 23, now on 101, now on -99. Say what you are seeing to
others. Try to see what they say.
F
Comments Many people do not naturally imagine the line stretched out horizontally from left to right.
For some the line is jagged or curved, for some it rises upwards, and these are only a few of the variations.
Comparing notes on different ways of seeing the number-line can be instructive, but it is much easier to
communicate mathematically if people can also imagine a 'standard' left-to-rightstraight number-line when
dealing with mathematical questions.
MOVING ABOUT ON THE INTEGERS Which integer is three to the right of 5? Which integer is
six to the left of 4? Which integer is twice as far from 3 a s from -7? Half a s far from 9 as from
5? Go round the group, each person suggesting a question for the others.
b
Comments The purpose of this activity is to gain confidence and familiarity with a sense of the integers
stretched out on the line.
WHAT IS A NUMBER? The above two activities assumed acceptance of the idea that 0 and -2
are both numbers. Just what is a number? Do your pupils accept that 0 and -2 are numbers?
What is required to induce someone to be flexible in their use of a mathematical term such as
number?
F
Comments Everyone is familiar with names of numbers, but mathematicians have shifted over the
centuries in what they take a number to be. As with most definitions of mathematical terms, the precise
meaning of number depends on the context and purpose for its use. Thus, number can be interpreted as
'natural number', meaning the counting numbers 1,2, 3, . . ., but it can also be taken to include negatives,
fractions, decimals and more. It is convenient, however, to extend the counting numbers to larger and larger
systems of number-like entities and then to consider these extended systems also to be numbers.
FINITE DECIMAL-NAMES
Decimal number-names such as 1.24 or 3.14159265359 a r e called finite i n order to distinguish
t h e m from names i n which t h e digits keep going on a n d on forever, as i n 0.3, 42, or x . T h e
significance a n d meaning of infinite n a m e s is explored later.
b
WHERE ARE THE FINITE DECIMALS? Imagine t h e number-line again, w i t h t h e integers
marked on it. Now focus on the point half-way between 1 a n d 2. W h a t i s i t s decimal-name?
Focus also on the point whose decimal-name is 1.6. Where is it on the number-line? How f a r
a p a r t are 1.5 a n d 1.6? What is t h e name of t h e point half-way between 1.5 a n d 1.6? W h a t about
the point half-way between 1.55 and 1.56?
b
Comments Some pupils offer 1.5.5 for the name of the point half-way between 1.5 and 1.6. One
explanation is that they see the decimal point a s an indicator of 'next smaller stage', rather than as the
demarcation between whole numbers and (decimal) parts of a whole. They are in very good company,
because the first known printed appearance of decimals, in 1585, put a label in front of each decimal digit, to
indicate in which decimal place it belonged.
I t i s important to be able to read decimal-names, but there a r e sometimes several readings
used i n society, not all of which are mathematically helpful!
READING Some people read 0.16 as 'zero point sixteen'. I n what contexts i s i t commonly read
in this way? Use this reading to explain why some pupils might conclude t h a t :
since 0.5 X 0.5 = 0.25 and 0.4 x 0 . 4 = 0.16, 0.3 x 0 . 3 i s 0.9
1.23 i s bigger than 1.5
3.25 + 4.1 = 7.26
Predict what the same pupils might give as answers to the questions:
W h a t a r e the next two terms in the sequence 1.2, 1.4, 1.6, 1.8, . . . ?
What a r e the next two terms in t h e sequence 1.97, 1.98, 1.99, . . . ?
Where on a scale marked i n tenths would you locate the numbers l.5, 1.05 a n d 1.50?
W h a t would they do i n response to t h e t a s k of putting t h e following numbers i n increasing
order of size:
2.5, 1.35,2.45, 1.006, 1.07, 1.8?
b
Comments Some texts us a '. ' in the numbering of sections or paragraphs, and 1.10 would be expected to
follow after 1.9. In what other contexts is this sort of numbering system used? Often it helps pupils to
distinguish contexts correctly if they have been asked to consider for themselves in which contexts different
forms are used. For example, the symbol 171b , sighted a t a petrol station, suggests a slightly different
notation for decimal numbers.
Radio and television are influential sources of misread numbers, particularly decimals. They are misread in
the sense that language patterns such as 'zero point sixteen' lead pupils quite naturally into mathematical
errors and confusions. To conclude that 0.3 X 0.3 is 0.9 exhibits excellent mathematical thinking, misled by
an unfortunate non-mathematical reading. It is reasonable to assume that when pupils make mistakes, they
are doing it for good reasons, and not just to be wilful. What other errors are you aware of with decimal
numbers, and how might they be related to misnaming? See also The meaning and use of decimals by
M. Swan, Shell Centre, Nottingham, 1983, and Children's understanding of mathematics: l1- 16, ed. K. Hart,
John Murray, 1979.
Each of the next two activities is in several parts. Each part provides just one or two examples
of a type of question to be investigated, explored and generalised until you are thoroughly
confident about what i t is saying. The parts connect together to illustrate important features
about the naming of points on the number-line.
NAMING POINTS What is the decimal-name of the point half-way between 1.5 and 1.6?
Where is the point 1.56 on the number-line? What is the name of the decimal point half-way
between 1.55 and 1.56? Repeat this construction several times, in order to get a sense of the
process going on and on and on. Try to be precise and economical in how you say where a
point is to be found.
What is the decimal-name of the point one-tenth of the way from 1to 2? What is the decimalname of the point one-tenth of the way from 1.1to 1.2? What is the decimal-name of the point
one-tenth of the way between 1.11and 1.12?
What is the decimal-name of the point one-tenth to the right of l ? What is the decimal-name of
the point one-hundredth (strictly speaking, one one-hundredth) to the right of that? What is the
decimal-name of the point one-thousandth to the right of that? Keep going until you have
established a pattern and can explain it to colleagues.
P
Comments What similarities and differences are there between the second and third parts of this
activity?
) NAMING POINTS (harder) What is the decimal-name of the point three-tenths of the way from
1to 2? What is the name of the point thirty-three-hundredths of the way from 1to 2?
What is the decimal-name of the point one-tenth of the way from 1 to 2? What is the decimalname of the point one-tenth of the way from 1to that point? What is the decimal-name of the
point one-tenth of the way from 1to that point? Keep going until you have established a pattern
and can explain it to colleagues.
What is the decimal-name of the point half-way between 1 and 1.5? What is the name of the
point half-way between 1and that point? Keep going.
What differences are there between the second and third parts of this activity?
Comments When using the decimal notation it is much easier to answer questions about tenths-of-the-
way between numbers than about half-way between numbers.
REFLECTING What general idea is exemplified in each part of the above two activities? How
are the different parts related? What sense of finite decimals emerges?
P
Comments You might want to pay particular attention to the inner images that you have, the kinds of
words and phrases (language patterns) used-to talk about those images, and the techniques for reading or
using finite decimals which are implicit in the activities.
Until now, activities have been about naming points which are on the number-line. The
reverse process is also important: locating points which correspond to decimal numbernames.
LOCATING FINITE DECIMALS Where would you find the following points on the number-line?
5, 5., 5.0 and 05.0
.7,0.7 and 0.70
Comments The decimal system of naming allows several different names for the same number. You do
not often see 5. written, but strictly speaking, 5. is the decimal-name for the number five, while 5 is not a
decimal-name at all, but rather a whole-number-name. As names, they are different, so if they are seen as
numbers, it is logical to assume that they are different numbers. The emphasis on number-names is
deliberate, because there is potential confusion when 5 and 5.0 are thought of as numbers, rather than as
number-names.
LOCATING MORE FINITE DECIMALS Where would you find the following points on the numberline?
1.7 and 1.8
1.73 and 1.74
1.732 and 1.733
1.7320 and 1.7321
How far apart are each pair? Pay particular attention to stating how each point can be labelled,
and being a s economical a s possible in your instructions. Square each of the numbers using a
calculator. Explain to a colleague how to extend the sequence of pairs to preserve the pattern
t h a t emerges.
Comments Note that the language has slipped back to speaking as if 1.73 was a number, rather than a
number-name. Emphasis is being placed on the use of language because of the close links between the way
people talk about mathematical ideas, and the way that they think about them.
REFLECTING What is special about the interval between 1 and 2 in locating and naming
decimals? What aspects of decimal-names for points emerged for you?
Comments Mathematics is deeply imbued with expressing generality, yet it is easy to overlook the
general when attending to the particular (see PM751 EXPRESSING GENERALITY, also in the MATHEMATICS
UPDATE series).
WHERE ARE THE DECIMALS (again)? Imagine the number-line extending indefinitely to the
left a n d to the right. Pick a decimal number-name like 23.45, with a t least four digits and a
decimal point somewhere in it. What is the label of the point six to the left, four to the right, ten
times a s far from 0, one-tenth of the way to zero, three-tenths to the left, two one-hundredths to
the right?
Comments The point of this activity is to achieve confidence with the labelling of points on the line and
with the points associated with similar labels. Questions about the arithmetic of decimal numbers are taken
up in Section 5.
When introducing a new topic to pupils, i t is tempting to explain things in detail, in the hope
t h a t they will understand rather t h a n simply learn to mimic t h e required behaviour.
Unfortunately, untimely explanation can often be treated a s wallpaper, a s simply something
else in the background to be coped with. Alternatively, pupils can be offered entry points to a
topic and then helped to use their powers of making sense to work out the details. That is the
general aim of all the activities in DEALING WITH DECIMALS, and particularly the focus of the
next one.
MINIMUM INFORMATION What is the minimum information of the form 'another way to
write 1/10 i s 0.1' in order to equip someone to work out (through discussion with colleagues)
how to use a n d interpret correctly (finite) decimal number-names? Typical challenges to
prompt deductions might include:
What is the decimal-name for 1/10, 2/10, . . .?
What are the decimal-names for ll2, ll4, 1/100, 10/1000,67/100,213/1000, . . . ?
What is the significance or meaning of each digit in the number-name 987.65432?
Comments Would pupils in such a situation think of names such as 00.1, or 0.10, and would there be any
difficulty deciding on their meaning? Do they need to be told a name for 3/10? Do they need to be told a
decimal-name for 1/100? In a mathematical atmosphere, they can make conjectures and modify them in the
light of experience, and remarkably little needs to be explained about naming decimals!
F
BETWEEN Find two (finite) decimal number-names for points lying between 1 and 2. Now
find two more between them. Now find two more between them. Outline a strategy for
carrying on indefinitely!
F
Comments A significant part of this activity lies in the care and precision with which you can outline
your strategy to someone else, so that they know what you are talking about!
F
REFLECTING Link your responses to BETWEEN with your work on LOCATING MORE FINITE
DECIMALS (see p. 10).
F
STORIES A good way to discover how pupils are seeing or thinking about numbers is to ask
them to make up a story about a situation in which the calculation 4 + 5 = 9 is relevant, and
another in which 1.3 + 2.6 = 3.9 i s relevant. What differences would you expect to find in
stories constructed about the two kinds of numbers? What forms of language might reveal how
pupils are thinking about the numbers?
Comments The best way to consider this question is to try constructing your own stories, and then to ask
pupils to construct some. You might like to consider using other arithmetical operations as well (see
GLOSSARY).
INFINITE DECIMAL-NAMES
This subsection looks at t h e locating of points on t h e number-line which have infinite
decimal-names. Since infinite decimal-names are concerned with fractions and irrational
numbers, i t might be worth looking in Section 2 a t the subsection on Fractions, either before or
while working on this subsection.
F
WHERE ARE THE INFINITE DECIMALS? Imagine the number-line a s before. Imagine also the
point which is one-third of the way between 1 and 2. What decimal-name does i t have? Do the
same for the point two-thirds of the way between 1and 2.
F
Comments The most important feature of decimal-names is that some of the names are infinitely long!
The name of the point one-third of the way from 1 to 2 is 1.33333. . ., where the dots indicate that there is an
infinite number of 3s present in the name. A more compact and standard notation is 1.3,which is read as
'one-point-three-recurring'.
F
WHERE ARE THE INFINITE DECIMALS? (continued) The decimal-name 0.3 labels a point onethird of the way from 0 to 1. Use this fact alone to describe t h e locations of the points with
decimal-names: 0.13,0.23,0.113,0.123, 0.213,0.223,0.313and 0.323
.
Generalise.
Comments By treating the symbol 0.3 as a name for a proportion or fraction of a distance, you can
describe the position of each of the points. By treating 0.3 as another name for 1/3, you can work out what
fraction is represented by each of the proposed names.
REFLECTING What is the difference, if any, in thinking of 0.3 as a number, and a s a numbername?
Comments It is common for pupils to keep fractions and decimals in different compartments, and most
pupils have difficulty in conceiving of 'a number that goes on forever'. Recurring decimals arise in more
detail in Section 2. In PM752 APPROACHING INFINITY, also in the MATHEMATICS UPDATE series, it is
stressed that infinite decimal-namesare names for bonafide numbers, and that as well as seeing the name as
going on forever, it is also possible, and useful, to see it as a completed act or process.
F
LOCATING INFINITE DECIMAL-NAMES Let w (for whole) denote the decimal-name
(where the spaces are simply to help reveal the structure). How would you instruct someone t o
locate a point on the number-line corresponding to W ? Don't be too concerned with the details of
W ,but rather, use i t a s a particular example to illustrate a general method of locating a point
corresponding t o a n infinite decimal-name.
Comments Locating W exactly is impossible, but it is technically difficult to justify this statement, even
after you have been precise about what is meant by locating a point exactly. The only strategy available is to
get closer and closer to the required point, even if you can't 'point it out' exactly. You can trap it inside
smaller and smaller intervals of width 1/1000,1/10000, etc.
BETWEEN (again) Find two distinct numbers (both with infinite decimal-names) lying
between 1 and 2. Now find two numbers lying between them, with finite decimal-names. Now
find two numbers lying between them with infinite decimal-names. Keep going, alternating
finite and infinite names, until you can convince a colleague t h a t you have a method for doing
both tasks in general.
Comments -You may run into difficulty deciding whether two different names represent the same or
different numbers. This question is taken up in the next section. The idea lying behind this activity can be
exploited to find solutions to difficult mathematical questions, as well as to enable mathematicians to be
much more precise about what is meant by decimal number (see Section 4). The theme of BETWEEN is taken
up again in Section 3.
BETWEEN (yet again) Suppose someone h a s been trying to locate a decimal-name on t h e
number-line, and they have trapped i t between the points labelled 12.345 a n d 12.346. What
possible values could the number actually have? What could it mean to 'know the value correct
to two decimal places'?
F
Comments There are at least two possible meanings for 'correct to two decimal places' (see rounding and
truncating in the GLOSSARY). ?ky to specify a precise method for rounding and for truncating to a given
number of decimal places.
b
EXTRA DIGITS Most calculators show only some of the digits of a number (usually eight). How
might you find out how many digits are hidden? Try finding how many digits of x are held by
your calculator. The value to sixteen places i s 3.1415926535897932 for comparison. Find an
efficient way to explain your technique to a colleague.
Work out a method for entering a number in your calculator which h a s the maximum number
of digits i t can hold (not just receive directly).
b
CALCULATOR ROUNDING Use a cal&lator to find out whether the last digit held in the display
i s correct in each of the following:
divide l by 7
divide l by 3
divide l by 6
Experiment with other calculations to find out how and when your calculator rounds or
truncates.
Predict what will happen if you enter:
0.3333 . . .3d (where the last digit d is a 1 , 2 or 3) and then multiply by 3
0.6666 . . . 6d (where the last digit d is a 6 or a 7) and then multiply by 6
Comments Your theory about what your calculator does about rounding or truncating should enable you
to decide what it will do here.
REFLECTING Specify how you would go about locating a point on t h e number-line .
corresponding to a number with a n infinite decimal-name. Pay particular attention to the
technical terms and expressions which you need.
b
The decimal-names for points on the number-line seem perfectly straightforward. To locate a
point with a particular name approximately, you simply use a s many decimal places a s are
required, and then find the requisite point on the number-line. But what about 0.9 ?
t
RECURRING NINES The sequence of decimal numbers 0.9, 0.99, 0.999, 0.9999,
. . . gets closer
1
and closer to a t least one decimal number, namely 0.9 . What other 'well-known' numbeds)
do the members of the sequence get closer and closer to?
b
Comments Strictly speaking, the members of the sequence get closer and closer to any decimal number
which you care to name (as long as it is at least as large as l), for example, 100.83. The reason is that the
members of the sequence get larger and larger, and so get closer to 100.83. You might wish to argue that the
expression 'gets closer and closer to' means that the members of the sequence get very close, which they
certainly do not do for 100.83. But do the members of the sequence get very close to 1.0000000001? In fact.
they never get closer than 0.0000000001, so if we are looking extremely closely, the sequence members do
not get very close even to 1.0000000001.
The members of the sequence 0.9, 0.99, 0.999, 0.9999, . . . are all finite decimal-names. As
names for points on the number-line, they name points which approach a limiting position,
which i s also a point on the number-line. The name of the limiting value is 0.9 (from looking
a t the names), and its name i s 1 (from looking a t its position on the number-line): There can
be no point between the points labelled by 0.9 and 1, since there i s no decimal-name available
for such a point. P u t another way, the strategy you developed for locating a point does not yield
a possible decimal number-name here, so there cannot be any such point. (The pack PM752C
APPROACHING INFINITY, also in the MATHEMATICS UPDATE series, looks in more detail a t this
issue.)
EXTENDING In NAMING POINTS (see p. g), you were asked to locate numbers obtained by
successively adding one-tenth, one-hundredth, one-thousandth and so on. Name some
decimal numbers which are bigger than all of the terms in this sequence, and to which your
sequence of numbers gets closer and closer. Such a number is called an upper bound for the set
of numbers being considered. What is the smallest of these upper bound numbers?
Do the same for the sequences generated by:
one-tenth from 1to 2, one-tenth from there to 1.2, one-tenth from there to 1.12, . . .
one-tenth from 1to 2, one-tenth from there to 2, one-tenth from there to 2, . . .
one-tenth from 1to 2, one-tenth from there to 2, one-tenth from there to 1.2, one-tenth
from there to 1.2, one-tenth from there to 1.12, one-tenth from there to 1.12, . . .
Comments The first sequence never gets larger than 2, never gets larger than 1.2, than 1.12,1.112, etc. It
gets closer and closer to all of these, but by the phrase 'gets closer and closer to' mathematicians usually
mean that the sequence gets as close as you wish. The first sequence cannot get closer than 0.001 to 1.2, but it
1
can get as close as you wish to 1;. Thus any of the numbers mentioned are upper bounds, but lgis the
smallest possible upper bound.
The second and third sequences are certainly bounded above by 2, but is there a smaller upper bound for
either of them? For the last sequence, be sure you are clear on what the 'dot dot dot' means, that is, how you
interpret the pattern continuing.
NINETEEN What is the least (smallest) upper bound for the sequence of numbers:
Comments These numbers all have finite decimal-names, but they get closer and closer to the number
,,
I
1
with decimal-name 19.19. Here the two dots indicate that the recurring part is '19'. Thus 19.19 means
19.19191919. . .,where the name consists of an infinite number of repetitions of '19'. This is also the least
upper bound of the last sequence in the previous activity.
ABSTRACTION
The point of this section was to draw attention to the use of decimal-names for points on the
number-line, and to suggest two things:
every decimal-name refers to a point on the number-line;
every point on the number-line has a t least one decimal-name.
Stress was placed on the idea of a number-name rather than the number itself, in order to allow
for the possibility that the same number can have many different names. Some points on the
number-line have a unique decimal-name, while others have infinitely many different
decimal-names (counting tails of zeros of different lengths). For example, the number one
has decimal names l., 1.0, 1.00, . . . 1.0, as well as 0.9 . Any finite decimal-name can have a
string of zeros put a t the end. Even if the ambiguity about zeros is removed by demanding that
all decimal-names are infinite, there still remains the alternative of replacing the last nonzero digit by the digit one less followed by nine recurring. This multiplicity of names is
inescapable, and in fact is quite useful.
1
l
ACCURACY! The decimal-names 3, 3.0, 3.00, 3.000, . . . and even 3.0 all represent or name
the same number. Far from being a disadvantage, this use of 0 a t the end of finite decimalnames is used when making measurements, to indicate that the number being talked about is
an approximation. The number of zeros shown a t the end is used to draw attention to the
number of decimal places of accuracy claimed in the measurement. Why might such a
1
convention be needed? What then is the meaning of 3.0 a s a measurement?
F
Comments It is not usual to use the name 3.0 since this would indicate absolute precision, and the name 3
without a decimal point is much more succinct and familiar as a name for the same number.
F
UNIQUENESS OF NAMES Focussing only on the use of recurring nines (and ignoring the use of
zeros front and back), which pairs of decimal-names are actually names for the same
number? Try to describe all such pairs. Which points (if any) on the number-line have a
single, unique name, and which have two or more names? How many decimal-names do they
have? Formulate a general rule.
F
Comments Does your rule apply to zero? There is no escaping having two names for the same decimal
number, at least for some numbers. It is possible to arrange that different numbers have different names, but
not that there be a unique name for each number. Consequently, it is often convenient to agree that
whenever decimal-names are being spoken about, names with a tail of repeating nines are banned. In other
words, if there is a finite decimal-name, use it! In Section 4, however, it is shown that nine recurring as part
of a name cannot be banned altogether, because it sometimes arises in arithmetic calculations.
To find the decimal-name of a given point, you may have to be satisfied with knowing only
part of the name (that is, knowing the number approximately), found by trapping the point
between closer and closer pairs of points with known names.
The set of all numbers corresponding to points on the number-line are called real numbers
(see GLOSSARY). The only reason for using the name 'real' is to distinguish them from later
inventions of non-real numbers which lie in the number-plane but not on the number-line.
But that is a more complex story!
REFLECTING In what way has your sense of the number-line and the naming of points on i t
changed or developed as a result of working on this section?
F
F
REFLECTING FURTHER This pack is a form of mathematics text. The nature of text is that
choices have to be made as to order, emphasis and style of introduction of ideas and activities.
What choices can you detect having been made in this section? What choices might you make
for working on the same ideas with your pupils?
Comments The series PM753, with titles involving the phrase 'PREPARING TO TEACH . . .', uses a sixfold
framework which is helpful in reflecting on choices in presentation, order, particular activities, and emphasis
as the lesson proceeds. The framework could also be used to analyse the presentation of this section.
F
ASSESSING What behaviour on the part of pupils might suggest that they have mastered the
labelling of points on the number-line with decimal-names, and the locating of points
corresponding to given decimal-names?
F
2 MULTIPLE NAMES
The previous section stressed the idea of names for numbers rather than direct contact with the
numbers themselves. One of the main reasons for this is that there are many different names
for the same 'number'. In this section, the theme of numbers having many names is
developed further, because when names like 2, 412, and 1 + 1are treated as numbers, there'is
potential for confusion, because the difference in name is sometimes taken as reflecting a
difference in value. The aim of the activities is to get a sense of the multitude of numbers, and
the multitude of number-names for the same value. There are short subsections on the more
familiar whole numbers and fractions, and a longer one on the less familiar irrationals.
WHOLE NUMBERS
P
THINK OF A NUMBER Think of a number between one and ten.
B
Comments Did you think of a whole number? Why? Why not 0.5, or 0.12354, or 4212, or
. ..?
Could it be
that pupils associate the word 'number' only with whole numbers because they rarely hear it used in any
other context? You might like to see how often you use the word 'number' when referring to decimals or
fractions, perhaps by getting someone (a pupil?) to keep track during a few lessons.
CREATIVE ARITHMETIC The symbol sequence '3 + 4 = ?' invokes closure onto a single answer,
namely 7. The sequence '7 = ? + ?' offers an opening to a sense of many possibilities. How
many possibilities are there?
F
Comments The question is intentionally vague. It all depends on whether you feel constrained to positive
whole numbers, or whether you countenance negative numbers, fractions or decimals.
F
ARITHMETIC NAMES FOR NUMBERS The number named by 7 can also be named by 3 + 4, or
1 + 1 + 1 + 1 + 3. How many different names for 7 can you make using just four arithmetic
operations? What if you restrict yourself to one use of each operation? Are there any properties
of 7 which make i t harder or easier to describe in this way than other numbers? Generalise!
Comments There is a tendency to see 3 + 4 as an uncompleted operation, but it is important for algebraic
thinking to be able to see 3 + 4 both as a calculation and as the answer to that calculation. The point of the
activities in this subsection is mostly to emphasise that there are many different ways to name numbers, and
that this is only the beginning (see also PM753C PREPARING TO TEACH EQUATIONS, also in the
MATHEMATICS UPDATE series, for a development of this idea).
F
REFLECTING What are the pros and cons of thinking of 3 + 4 as a number-name, as well as an
operation on number(- name)^?
ASSESSING What behaviour on the part of pupils might suggest that they appreciate the
multiplicity of arithmetic names for a whole number? What questions might you use to probe
for this behaviour?
FRACTIONS
Is halving the same a s taking three-sixths of something? The answer is yes, and no, because
the actions are different though the result (apart from possible crumbs) is usually the same.
Finding half of a collection of objects is an operation on those objects, and turning halving into
a number one-half is a sophisticated if often glossed over act. It is taken up in Section 3.
The symbol V3 is similar to 2 + 3 in that i t represents both a calculation to be performed and a
number-name for the answer to that calculation. Many pupils seem to find i t odd that 2/3 is
both a fraction and a calculation.
If fractions are accepted as names for points on the number-line, and hence for numbers, then
which decimals do they correspond to?
F
INTO DECIMALS Which fraction-names convert into finite decimals?
Comments The only way to make progress is to try some examples. The purpose of the examples i s to
use them to detect a pattern in those fractions which do, and those which do not, convert to finite decimalnames. Being systematic in the choice of examples i s sensible, but the systematicity arises a s part of the
pattern-spotting process (see the BOOKMARK). A calculator may be helpful to start with, but to provide a
convincing argument you may need to do some long division! In order to investigate the nature of the
decimal-names of fractions, i t helps to use the calculator efficiently (see EXTRA DIGITS on p. 13).
F
MORE INTO DECIMALS Convert ll9, 1/99 and 11999 into decimals. Although a calculator is
useful, it is also helpful to do the long division so that you can see what is happening.
Deduce the recurring decimal-names of the following fractions (there is no need to do any
long divisions for these once you know the decimal-names for 1/99 etc., though you might like
to check your deductions by hand or by calculator!):
lf33,
1/11,
2/33,
2/11,
41111
P
) RECURRING Writing down a periodic decimal-name (see GLOSSARY) is easy. For example,
0.87189189189. . ., which in its shorthand form is 0.87i89, is a typical periodic decimal, but
what fraction does it correspond to?
(Suggestion: split off the recurring part from the rest. Then generalise the computations from
MORE INTO DECIMALS.)
Comments The aim i s to work out a general method of converting periodic decimal-names into fractionnames, by experimenting and looking for patterns (see the BOOKMARK for more'suggestions).
) MORE RECURRING Use a calculator to find the recurring decimal-names of the following
fractions:
Explain any connections which emerge between the patterns.
Comments You might like to focus particularly on 91'9in the last sequence, and explain to a colleague why
the pattern suddenly changes there.
ALWAYS? Why is i t that when fraction-names are converted into decimal-names, the result
is always either finite or recurring?
Comments It may be helpful to think of a finite decimal-name as ending in zero recurring, so that all
fraction-names convert into recurring decimal-names. If you don't know how to start on such a general
question, specialise! Look back over the special cases you have worked on, and try to see why they always
end up being recurring. It may be helpful to carry out some long divisions for 117 and 1/13.
FOR THE BOLD Decide whether the following conjecture is true. If necessary, modify it until
i t is true. Then find a way to convince yourself and a colleague that you are correct.
Every integer divides evenly (no remainder) into a t least one (and hence into
infinitely many) of the numbers in the set (9,99,999,9999, . . .).
What other sets of numbers have similar properties?
F
Comments For example, is the same thing true for (8,88,888, . . .), or for (18, 1818, 181818, . . .)?
REFLECTING What are the pros and cons of the theme of multiple names in the context of
fractions?
F
ASSESSING What behaviour on the part of pupils might suggest that they appreciate the
multiplicity of fraction-names for a number? What questions might you use to probe for this
behaviour?
F
IRRATIONALS
Numbers which correspond to positions on the number-line are called real numbers. Those
which are not rational (that is, numbers which do not have fraction-names) are called
irrational, meaning literally 'not-rational', to indicate that they cannot be expressed a s a
ratio of two whole numbers. Thus any number which is not a fraction is non-rational or
irrational. The association of reasonableness with 'rational' derives from the Pythagorean
world view in which all aspects of the world, both physical and psychical, are based on simple
ratios of whole numbers. Unreasonableness, which is associated with the word 'irrational',
stems from the same source, though whether the Pythagoreans deserve the blame is open to
doubt.
The aim of this subsection is to show that there are many irrational numbers whose decimalnames can be thought of, if not written down completely, but that these are only a tiny part of the
irrational numbers. The first activities take up the theme that every point on the number-line
has an associated number, and hence a decimal number-name as well as other possible
names. The activities vary considerably in complexity, so look for something which you find
attractive, and don't be put off by the occasional flurry of symbols. Whatever activities you
choose to explore, be sure to look a t the last subsection Abstraction.
42 AND OTHER SURDS
One of the early successes of geometry was a demonstration generally ascribed to Euclid that
42 is irrational, and his proof that certain numbers are irrational is of concern to
mathematicians who study the properties of numbers. His argument, and a modern version,
are examined in the subsection Surds for the bold.
DIAGONAL In the following diagrams, each row represents the first and last frames from an
animation, illustrating a geometric construction of a length on the number-line. In each case,
decide what name(s) you would give to the constructed point on the end of the diagonal of the
last frame.
What points can you construct on the number-line in this fashion, and what names are they
most often known by?
Comments To make a start, you need to decide what 'in this fashion' means. The distinction is between a
. . ., and the name of the point in the form of 45 etc. Don't
forget that you can use lengths already constructed in order to construct new ones. Could you use
something similar to locate the cube-root of 2? The fourth root?
point being constructed, described as the length of
F
TRAPPING Suppose you want to find as accurately as possible the decimal-name of a positive
number which has the property that it is one less than its square. You could write down an
equation and try to solve it, but you could also trap it in between two numbers, say 1and 2, by
observing that 1 is more than 'one less than its square', while 2 is less than 'one less than its
square'. By successively narrowing down the interval in which the number lies, you can get
closer and closer. An efficient way to do this is to examine the mid-point of the interval [l, 21,
and decide which half still traps the number you seek. Use your calculator to do this, correct to
four decimal places.
F
Comments The number for which you have just found a n approximate value is known a s the golden
l+&
ratio, with names 2 and 7 .
It occurs in many unexpected places in mathematics and nature, from the
famous Fibonacci sequence to sunflower seed patterns, from diagonals of pentagons to the proportions of
the Parthenon and even to the shape of the new television screens. The trapping approach is often called the
bisection method because a t each stage you cut down the error interval (the length of the interval in which
the number is trapped) by one-half (see GLOSSARY). It is a very general and powerful technique. I t is
invoked quite naturally by eye-hand coordination, and when cutting something to fit or fill a gap. (The pack
PM753C PREPARING TO TEACH EQUATIONS, also in the MATHEMATICS UPDATE series, explores in more
detail the bisection method of solving equations.)
REFLECTING Describe to yourself or to a colleague how the idea behind TRAPPING could be used
in other situations. For example, describe how the idea can be used to find a number which is
one less than its cube, or its fifth power, or its one-hundreth power, or to find an angle 8 for
which sin(8) = 8.
Comments I t may be worthwhile writing a short computer program to carry out trapping in a variety of
different settings, a s a handy tool for investigating numbers.
The most famous irrational number, apart from 42, must be the number whose short name is
pi, or n (a symbol first used by Euler in the 18th century). It is both the ratio of the circumference
of a circle to its diameter, and the ratio of the area of a circle to the square of its radius. These
two definitions of R both depend on the fact that the ratio in question is independent of the size of
the circle, but it is not at all obvious that these two ratios should be the same.
TWO SOURCES FOR PI Something similar to the following diagram appears in many
textbook treatments of R. Interpret the diagram as the basis for an argument that the ratio of the
circumference of a circle to the diameter is the same as the ratio of the area to the square of the
radius.
The value of R is usually estimated by measuring the circumference and diameter of various
circles, but you can only get one or two decimal places of accuracy. The well-known
20
approximations 2217 and 3551113 were derived by Archimedes (and probably others), and
displaced the ancient approximations of 2518 (Babylonian) and 256181 (Egyptian) which were
in use around 2000 BC. Curiously, 6 1 - 4 is approximately equal to' n: a s well (see
APPROXIMATIONS TO PI on p. 23 for more such approximations). To show that x is not 2217,
nor 3551113, nor 3.141, nor 3.1412, you have to take larger and larger regular polygons
approximating to a circle. The essence of the idea is again that of trapping.
F
POLYGONAL APPROXIMATIONS Use the following pictures to construct a story which suggests
how one might go about trapping the value of x.
Comments The story you constructed was the method developed by Archimedes for trapping the value
of R.
F
CALCULATOR PI A calculator offers many more places for X . Enter the value of x given by
your calculator. Write down the answer, then multiply the answer by 2, both on paper and
using the calculator. How do you account for what happens?
Calculators actually store more decimal places than they show (see EXTRA DIGITS in Section
1,p. 13for more details). The full value stored by any calculator is not, however, the full value of R,since the
decimal-name of x is non-terminatingand non-recurring. Only the first few hundred million decimal places
are known!
Comments
The following activity provides an approach to trapping the value of TC without the use of
trigonometry. Archimedes' method is, of course, fine in principle, but it is difficult to carry
out very far. The approach which follows carries out Archimedes' idea in a special case, but
requires familiarity with Pythagoras and with manipulating symbols. If you are' not
interested, omit it and look a t APPROACHES TO PI, which follows.
TRAPPING PI CONTINUED In 1592, Francois Vieta used Archimedes' idea by doubling the
number of edges in the polygon each time, to get the first known formula for X, from which can
be obtained a sequence of numbers involving only square roots which converges to n:. Use the
following diagrams and remarks to reconstruct his argument. The diagrams are intended to
indicate how to move from the kth av~roximationto the (k+l)th. The third is an enlarged
version so that marked lengths can be-seen.
From the diagrams, use Pythagoras' theorem to deduce that
and
Use these to deduce that
2ck+12= l + Ck
and that
-- 1
4
%+l
%+l
This enables successive values of ck to be found, starting with cl =
values are
G,so that the successive
Since the value of n: is approximated by the perimeter of the 2k-sided polygon with edge-length
ak, and radius 1, and since
-----1 1
Ch -1
-l
.
.
.
-2 C h -3
c1
1
1
Vieta's formula for approximating n: emerges as
Comments The purpose of presenting these algebraic calculations is to offer one route to a sequence of
calculations for calculating K in practice. There are several other approaches, but the more efficient they are,
the more complicated is their derivation.
APPROACHES TO PI Use a calculator to find the first few terms of each of the following
sequences, and discuss their use as approximations to the value of X .
Leibniz (1673):
Euler (1707-1783):
Comments All of these sequences do actually converge to K, though fairly slowly. Proving that this is the
case is quite difficult.
APPROXIMATIONS TO PI Each of the following expressions h a s been used a s an
approximation to X a t some time. Use a calculator to find out how accurate they are.
Archimedes (225 BC):
2118%
67441
Ptolemy (AD 150):
377
m
Tsu Ch'ung-chin (AD 430501):
E
355
Lambert (1728-1777):
Ramanujan (1887-1920):
(Try finding more like these last ones!)
--
P
22
365
103993
Comments The values T , E,and m a r e the best possible fractional approximations using whole
numbers of those magnitudes in numerator and denominator (see D. Wells, The Penguin dictionary of
curious and interesting numbers, Penguin 1986).
REFLECTING What is the difference betiveen a sequence of approximations to a a s in
APPROACHES TO PI and approximate values of n as in the previous activity? What are the
implications of no-one knowing all of the decimal digits of X? In what sense is n really a
number?
-
.
IRRATIONAL DECIMALS
Irrationals are characterised by having no repeating tail in their decimal-name. That makes
i t easy to write down all sorts of irrationals; for example, the following decimals are
irrational (the spacing is provided between some digits simply to make the rule for formation
stand out):
F
MAKING IRRATIONALS Decide on a rule which will justify t h e . . . in these two decimalnames, and satisfy yourself that the rule does not generate any repeating tail. This is possibly
not quite a s easy a s i t sounds. Then make up a variety of your own rules and convince a
colleague that they generate irrational numbers.
I t is.possible to use decimal notation to make up numbers with surprising properties. Even
odder, a slight change in the specification of the number can change the properties completely.
THE NUMBER w Imagine writing down all the positive integers one after another to form the
number
W
(for whole):
(Again, the spaces are inserted in the decimal-name in order to make the formation of the
number-name clear.) What sort of a number is this? Does it have a finite or infinite decimalname? Is i t a rational or an irrational?
Can you predict the 100th digit or the nth digit?
Comments Deciding what sort of number it is is easy - since it is not repeating or finite, it must be
irrational. But writing down a formula for predicting the nth digit is not easy to do. It is much easier to say
in general terms how to go about finding it -just keep on writing!
The idea of putting consecutive numbers one after another as the digits of a decimal suggests many other
variations, but it is very easy to find yourself deeply involved in number theory. For example, the decimalname formed by writing the prime numbers one after another is intuitively irrational, but it is not at all easy
to show that it is actually periodic.
The number W is very useful, because it codes every possible whole number, and it can-be used
to encode sets of whole numbers a s well. Thus, the decimal number-name
when thought of alongside W , and where the . . . is taken to mean that there are no more zeros
in unexpected places, is a way of coding the set (6, 10, 13) as a decimal. Similarly,
when thought of alongside W, and where t h e . . . is taken to mean that the pattern of zeros
continues throughout the whole decimal-name, is a way of encoding the set of positive even
numbers as a decimal. The number W can then be used to show that the collection of all subsets
F
of the whole numbers, finite or infinite, can be matched one-to-one with a subset of the decimal
numbers (see PM752C APPROACHING INFINITY for more details of this sort of activity).
Now make just a little change. When writing down each of the whole numbers in turn, move
only one place to the right each time instead of giving each number its full space, so that there
are forward carries from whole numbers of two, three, four, . . . digits:
or in other words,
W = 0.1 + 0.02 + 0.003 + 0.0004 + . . .
to give
F
W = O . l 2 3 4 5 6 7 9 l ? ? ? ...
THE NUMBER W Work out the first 20 or so decimal places bf W. Can you be sure that your
digits are correct and that there will be no more changes in them due to more terms being
added on the end? Is W the same as 10/81, and can you be sure of your answer?
F
Comments You can work out exactly what number W is, by observing that if you multiply W by 10 and
then subtract W from the answer, you get a well-known recurring decimal.
THE NUMBERS fib AND FIB The number fib is specified as:
'
where each term in the Fibonacci sequence is used to form the digits of an infinite decimal
number-name, with zeros a s separators. Convince yourself that fib is irrational. Is there any
ambiguity in fib with Fibonacci terms which end in zero?
The number FIBl is specified as
or alternatively a s
so that each successive term in the Fibonacci sequence is written one decimal place (the
significance of the 1in the name) further to the right. Similarly, FIB2 is specified as
where each succeeding term is shifted two places to the right, and the three and higher digit
terms contribute to earlier digits of the final number.
Are FIBl and FIBz rational or irrational?
b
Comments Oberve that dividing FIB, by 10 and adding it to FIB, produces a number not too dissimilar to
FIB,! Finding the period is not at all easy. Try constructing other similar numbers and deciding whether
they are rational or irrational.
D O N ' T FORGET THE ABSTRACTION A T THE E N D O F THIS SECTION!
SURDS FOR THE BOLD
Euclid used an ingenious argument to show that 42 could not be a fraction. I t is the argument
that is given in all the textbooks which goes as follows:
Suppose d2 ,=
:
where p and q are integers which have NO common factors other
than one. Then p2 = 2q2. Since 2 is a divisor of the right-hand side, it also
divides the left, which is p2. Therefore 2 must divide p. But then there are two
factors of 2 on the left-hand side, and only one on the right, so 2 must divide q as
well, which contradicts the fact that p and q have no factors in common.
3
The same argument can be used to show that 43, & and other such numbers called surds (from
the Latin for dull-sounding, in other words, not having to do with harmonics and probably
because they were at first deemed absurd (see GLOSSARY)).
Generations of pupils have found the argument by contradiction used by Euclid confusing.
You start by assuming what you claim to be false (that 42 is a rational), and then end up
contradicting some aspect of your assumption (that 42 was expressed in simplest terms), and
from the contradiction deduce what you claim is true. I t is, however, an important method for
proving certain kinds of statements. Pierre Fermat found i t fascinating, and developed i t into
his method of infinite descent. Here is one example of its use which not only proves Euclid's
result, but which generalises considerably. I t makes use of the fact that if
(Pause and verify this!)
Fermat's argument then proceeds a s follows:
Suppose d?=
F, where p and g are integers and p is the smallest such positiue
integer possible. Then 42 =
4
=
3
=
P
29 - t p
P-@
for any integer value o f t . Now t
But this
can be chosen so that 0 < p - tq c q by taking t = integer part of .:
would contradict the fact that q was chosen as the smallest possible denominator
for a fraction to represent h.
) MAKING SENSE Try to make sense of this argument, and then generalise it to show that
is not a fraction. Compare the argument with a Euclid type of argument of the same result.
REFLECTING Why can the argument given not be applied to 6 6 to show that it is irrational?
Can you show that 42 + 43 is irrational? Generalise!
Comments Try isolating either 62 or 43 and then squaring, and applying facts about 42 and 43.
Generalise not just by putting in other integers for 2 and 3, but also by adding in yet more 4 signs (see Surd
Arithmetic on p. 37 in the next section for more about surds).
BOX TILING Tiling a rectangular tray of dimensions 13 by 17 is not a t all difficult if you use
tiles which are unit squares. But can other shapes of boxes always be filled with squares all the
same size? Suppose you have a box measuring 42 by 342 units. Can it be filled by square tiles
all the same size? What about a box measuring 42 by 5, or a box measuring 42 by 43, or 42 by
3 + 42? Convince someone!
P
Comments To show that it can be done, you need merely state the size of the square tiles, and show
arithmetically that they fit. To show that it is impossible, no matter how (mathematically)small the square
tiles, it is necessary to produce an argument. A standard way to proceed is to assume that it could be done,
and then to show that this leads to a deduction which conflicts with what is already known (in this case, the
nature of 42). It can actually be proved that it is impossible to tile a rectangle, the ratio of whose sides is
irrational, using rectangular tiles of different shapes all of which have the ratio of their two sides rational.
The proof is rather difficult.
ABSTRACTION
Not only do numbers have many different names, each useful in different contexts, but the
invention of a notation, such as decimal-names, enables mathematicians to think about and
study numbers which would otherwise be impossible to think about. The numbers W and FIB,
and many others constructed along the same lines, are cases in point. By contrast, numbers
3
are best thought about in their non-decimal names.
like rc, and the many surds like 42 and
The number 42 has only two properties: it is positive, and its square is 2. It can be manipulated
algebraically just bearing those facts in mind. The number n arises in many different
geometrical contexts, but no algebraic symbol analogous to 42 makes sense, so Euler used the
symbol n, and that has stuck. But there are many, many other irrational numbers, unnamed,
even unthought-of. If Wordsworth had been a mathematician, he might have written that full
many an irrational was born to bloom unseen.
The ubiquitous presence of the calculator makes it more important then ever before that pupils
think about the nature of numbers, and particularly the connections between fraction-names
and decimal-names. Because the calculator shows only part of the decimal-name, i t is
necessary to find some convincing reason for asserting that a fraction-name names the same
number a s fan infinite decimal-name.
REFLECTING The aim of this section was to gain a sense of an abundance of irrational
numbers. To what extent has this been achieved? What activity was most helpful in achieving
the sense of irrational numbers that you now have?
b
ASSESSING What behaviour on the part of pupils might suggest that they appreciate the
multiplicity of irrational numbers and how they differ from rationals? What questions might
you use to probe for this behaviour?
P
See A. Gardiner, Infinite processes: background to analysis, Springer-Verlag, New York 1982, for a much
more detailed account of recurring fractions and irrational numbers.
3 DECIMAL ARITHMETIC
Numbers are often divided into three categories: whole numbers, rationals and irrationals.
Rationals are easy to identify from their decimal-names, a s long a s someone has indicated
the repeating part, but if they arise from an addition or multiplication, they may not be so
obvious! The aim of this section is to explore the extension of arithmetic from whole numbers,
through fractions, to decimals. Starting with common misconceptions held by pupils about the
arithmetic of decimals, some difficulties are encountered with adding and multiplying
decimals; a retreat is made to fractions and the transition into rational numbers; then the
arithmetic of rationals ,is developed, and finally, suggestions are made a s to how to use that to
specify how arithmetic is done with infinite decimals. The entire story took some 2000 years to
develop!
WHAT IS THE PROBLEM?
Adding together two numbers is a basic arithmetic operation, as is multiplication. The idea of
what constitutes a number grew, historically, from whole numbers through fractions to
decimals (though the full story is immensely complex a s one might imagine). At each stage, a
new notation for numbers was suggested in order to make thinking or computation easier and,
with the notation, the new kinds of number came (slowly) to be accepted. Each time the old idea
had to be integrated into the new, and the new seen a s an extension of the old. Thus, from
whole numbers, the move to fractions is seen as encompassing the whole numbers and so the
arithmetic of fractions must reduce to or extend the arithmetic of whole numbers. Similarly,
since fractions can also be represented using decimal-names, the arithmetic of decimals must
reduce to or extend the arithmetic of fractions. This process turns out to be more problematic
than might be expected!
WHY NOT? Why is 1.5 + 2.5 not 3.101
Why is 0.3 X 0.2 not 0.6?
Why is 1.2 X 100 not 1.200?
Why is 3.0110 not 3?
Why is 3.05110 not 3.5?
Why is 1.23 + 3.4'not .157?
In each case, construct a plausible reason (based on some memorised but perhaps incorrectly
applied rule of thumb) for a pupil to believe that the answer given is correct.
Commenls There are usually sound, but perhaps misapplied, reasons for the ideas that pupils pick up!
Rules such as 'to divide by ten, drop a zero' make sense in context but can easily be misapplied if the rule is
not your own summary of what is already understood. (See also READING in Section 1,p. 8.)
MOVING DECIMAL POINTS Rules such a s the following are often found in textbooks:
to multiply 3 by ten, add a nought to get 30;
to multiply 3.45 by ten, move the decimal point one place to the right to get 34.5.
What connections are there between these two rules, and how might a pupil be expected to
encounter that connection?
A pupil, trying to apply the corresponding rule for division, said
You can't move the decimal point in 2.3 to the left two places because there is
nowhere for it to go!
Try to see the decimal point a s fixed, and the number-name moving past it. What differences
in perception, if any, are involved in seeing the decimal point a s moving, and the decimal
point a s f x e d but the digits a s moving, and how does this relate to multiplication by ten?
b
DECIMAL SUMS Perform the following additions, using them to generate a rule or technique
for adding two decimal numbers.
MORE DECIMAL SUMS Can you predict the length of the period of the sum of two periodic
decimals? The following sums might help - use a calculator and your knowledge of fractions
to see what is happening.
Comments The periodic decimals in this activity were chosen to illustrate several aspects of adding
periodic decimals that might otherivise be overlooked. The invitation is to produce a complete theory or
description of the addition of such decimal numbers. To predict the length of the period of the sum of two
periodic decimals, specialise systematically in order to see what is going on. (Question - can you have a
periodic decimal-name in which the period is infinite?) You can also convert to fractions and then convert
back again of course.
INFINITE DECIMAL SUMS The following sums show t h a t it is pretty hard to add two infinite
decimal number-names together, especially if you don't know all the digits!
Can you be sure to get the sums indicated correct to, say, seven decimal places?
b
Comments When faced with adding two decimal numbers that end in . . ., it is pretty hard to work out
how to add them unless there is a pattern implied in the dots. To add 42 and 43 using decimals is suddenly a
non-trivial task! In fact, the only way to do it is to determine in advance how many decimal places you want
to know, and then work out the components to at least that accuracy. But wait! How do you know there
won't be any carrying which alters the entry?
SAME AND DIFFERENT What i s the same, and what i s different, about the products in each of
the following rows, and between the rows?
three 2s,
thirty-four 12s,
three 20s, ,
thirty-four 1.2.5,
thirty 20s
three point four 1.2s
b
Comments What 'rules' are pupils expected to deduce from such patterns? How are they helped to see
connections between the rules?
When a decimal-name i s the only or maior name of a number available, multiplication of
decimals turns out to be much more complicated even than addition.
F
DECIMAL PRODUCTS Begin to carry out each of the following products by long multiplication,
in order to experience the uncertainties inherent in multiplying infinite decimal-names.
Use your knowledge of fractions to verify what is suggested by the calculator.
v ,
Using long muliplication, at what point in' the calculation of 1.2 x 0.81 can you be sure of
even the first digit in the product? Just finding the first non-zero digit in a product of two periodic decimals
is not always easy if you confine yourself to some rule for multiplying the decimal-names without
converting to fractions.
Comments
F
MORE DECIMAL PRODUCTS Use the following products, a s well a s ones of your own devising,
in order to seek a rule for multiplying two decimal-names without converting to other names
first. You may find i t helpful, however, to do the conversion in order to work out what the
answer should be!
Comments Finding the period of the product of two periodic decimals is a hard task, even if you work
with the fractions. Note that there are connections between the three products in the last line and the number
W in Section 2 (see p.25). Ideas for these two activities were taken from D. Fowler, '400 years of decimal
fractions',in Mathematics Teaching 110, pp. 20-2, and 111, pp. 30-1.
DEDUCTIONS Use a calculator to conjecture the intended answers to the following collection of
related calculations, and hence deduce other names for the numbers involved.
FIXED IN PLACE When you add decimals, the decimal points are lined up and remain fixed
throughout the computation. How might you lay out decimal multiplication so that the decimal
points are lined up and stay fixed throughout? What about division?
Comments There is no suggestion that such a layout should be taught to pupils, but by working through
what is required, it is likely that you will find yourself rehearsing what you understand by place-value.
REFLECTING What is the source of the uncertainty in adding a n d multiplying deeimalnames of numbers?
D
ASSESSING What behaviour on the part of pupils might suggest that they appreciate the
connections between decimal multiplication and whole number multiplication? What
questions might you use to probe for this behaviour?
FRACTIONS INTO RATIONALS
In the light of there being some unsuspected difficulties with decimal arithmetic, i t may be
wise to go back to the arithmetic of fractions. The theme of decimal arithmetic reappears later
in this section.
FRACTIONS AS OPERATORS
D
THREE-SEVENTHS Locating a point which is three-sevenths of the way from 1 to 2 is not
difficult in principle, depending of course on how accurately you wish to locate it. How might
you go about it?
D
Comments In Section 5 GEOMETRICAL ARITHMETIC, there are indications of how one can use ruler,
pencil and compasses to locate any fraction as accurately as the sharpness of the pencil will permit. At this
stage, however, it is enough to observe that dividing the interval up into seven equal parts and then marking
the point which is at the right-hand end of the third part, locates l;on a number-line.
The essence of the last activity lies in the fact that you can calculate three-sevenths of
something. In this case, you can calculate or locate three-sevenths of the way from 1 to 2 by
dividing up the interval from 1 to 2 into seven equal parts, and then taking three of them. The
move to regarding three-sevenths as a number, divorced from any context of calculating
three-sevenths of something, is subtle and sophisticated. Pupils often do not immediately see a
link between, for example, 0.5 X 28 (which they feel they cannot do) and half of 28 (about which
they are confident). The number-line is useful as the context in which to move from threesevenths a s an operator (acting on intervals, but particularly on the interval 0 to 1 (see
GLOSSARY)), to three-sevenths a s the name of the point which is three-sevenths of the way from
0 to 1, to three-sevenths as the number-name for that point.
In other words, three-sevenths-of starts off a s an operator which operates on quantities (in this
case intervals) to give a result. I t then becomes a label for a point, and then a number. Long
before this transition to number is made, the arithmetic of fractions as operators is available to
pupils. It is a short step to move from fractions as operators to considering what happens if you
calculate three-sevenths of two-fifths of something, for example. It is not such a short step to
interpret this a s deserving of the name multiplication, nor is i t such a short step to then treat
these operators as numbers. The next three activities are concerned with these transitions.
) COPIES If taking one-seventh of an interval or other whole means dividing the whole up into
seven equal parts and taking one of them, then taking three-sevenths of a whole conforms with
the way we normally speak about taking 'copies of. Thus 'copies of is an expression or
language pattern associated with multiplication. What other expressions correspond to
multiplication?
Comments The root meaning of multiply is found in the word 'multiples'. There are also other related
expressions, such as 'repeated addition'; 'how many times?'; 'product of. In some sense, multiplication is
what is common to all the situations in which the various words are used, and in some sense it is amazing
that the mathematically abstract notion of multiplication should find application in so many different
situations. Notice also that we speak of three-sevenths as a plural, suggesting that it is three copies of oneseventh.
) MORE COPIES In COPIES, you were asked to consider three-sevenths as three copies of oneseventh. What then are one-half of two-sevenths and one-third of three-sevenths in terms of
'copies of?
F
COMPOUND OFS Thinking in terms of a n interval on the number-line, or of some other
object which can be split into parts of a whole, consider the following expressions, not
individually, but as samples from a sequence of bolder and bolder expressions whose values
can be found by extending values of earlier ones. Be careful to express your answers in terms
of parts of a whole.
one-third of one-fifth of a whole,
one-quarter of one-seventh of a whole,
one-third of two-fifths of a whole,
two-thirds of one-fifth of a whole
two-quarters of one-seventh of a whole
two-thirds of two-fifths of a whole
Compare your image of what you are doing with any patterns in your answers.
Comments This activity is posed using words in order to emphasise the operator role of the numbernames. It is useful when looking for patterns to use ordinary number-names. Notice that there is a sense of
dividing up a whole into parts, and there is a number pattern to do with tops and bottoms which provides an
equivalent 'single' operation.
When points on the number-line are identified with fractions a s operators on a unit interval,
t h a t is, a s labels which arise from carrying out a n operation, we get a typical mathematical
movement from process to process-as-object. The process names like 'one-third-of (the
interval from zero)' become point labels (one-third) and then become numbers by extension
from whole numbers. The arithmetic t h a t whole numbers enjoy gets extended to fractions.
Before this can be done confidently, however, i t is necessary to be more rigorous about how
fractions relate to whole numbers. For the moment, act a s if you accept t h a t compounded 'ofing' does indeed correspond to multiplication.
F
EXTENDING What language patterns could be used to find a connection between
multiplication by a whole number like 3 and 'of-ing' by 3/1?
F
Comments 'Three copies of, or 'three lots of express multiplication by three, and both can be shortened to
'three of. It is the sort of language you might hear in a fast-food restaurant, for example. You would
probably never hear anyone say 'three-oneths of something, but the operation of 'three-oneths-of-ing'means
taking three copies of what happens when you divide a whole into one equal part, which has the same effect
of course as taking three copies of the whole. So it seems reasonable to identify 311 with the whole number 3.
I t is not a t all obvious that 311 is the same a s 3,nor that arithmetic of fractions will extend the
arithmetic of whole numbers, so that, for example, 311 + 211 = 511. Many pupils place whole
numbers, fractions and decimals in different compartments, and do not think of them as
extensions which preserve the arithmetic of the smaller systems.
F
DIVISION The following questions are indicators of a string of questions which can be used
(preferably at a brisk pace) to draw attention to the fact that division of fractions is embedded
in the language of fractions and of division, and is within reach of all pupils. Each sequence
should be extended until everyone is answering fluently.
How many halves are in one? In two? In three? . . . In thump? In banana? In blog?
How many thirds are in one? In two? In three? . . . In seven? In thump? In blog?
How many quarters are i n . . .
...
What is the general pattern? (It should have emerged before you got to this point!)
P
MORE DIVISION What general patterns are present in the following questions? (One useful
way to tackle them is to get someone to read them out, while everyone else answers together, as
quickly as possible, so that a steady pace is maintained which pushes participants to keep up.
Each sequence should be continued until the answers are coming fluently.)
How many one-thirds are in one? How many two-thirds are in two? In three? . . .
How many two-thirds are in two? In two twos? In two threes? . . . In two thumps? In
two bananas? In blog? In seven? In eight? In nine? . . .
How many two-fifths are in two? In two twos? In two threes?
two bananas? In blog? In seven? In eight? In nine?. . .
. . . In two thumps?
How many three-fifths are in three? In three twos? In three threes?.
thumps? In three bananas? In blog? In seven? In eight? In nine?. . .
. . In
In
three
b
REFLECTING The claim has been made that a significant part of division of fractions is
syntactic (it reflects the structure of our language) rather than semantic (based on meaning
derived from different contexts). Prepare your case on this issue, and discuss it with
colleagues.
DOES ORDER COUNT? Why is two-thirds of five-sevenths of something the same as twosevenths of five-thirds of the same thing? Don't be satisfied with a quick or simple answer.
Discuss it with colleagues and try to be sceptical of any proposal until it has been cogently and
convincingly justified.
P
Comments Justification can be based on authority ('those are the rules') but this begs the question of
where the rules come from. It can also be based on primitive experience with objects (via 'copies of), but this
begs the question of why it always works.
REFLECTING What aspects of your sense of fractions have been supported, challenged,
amplified or extended by these activities?
b
This subsection focusses on fractions, but only because the usual way t o think of decimal
numbers (numbers with decimal-names) is as trapped between sequences of fractions. Before
looking a t the relation between fractions and decimals, it is important to consider the question
of why it is that the arithmetic operations (which began life applied to whole numbers) actually
work when carried over to or extended to fractions.
ARITHMETIC OF FRACTIONS
What does it mean to say that two fractions are equal? In some contexts, it makes sense, as
when taking three-sixths of a whole and one-half of a whole. But scoring three out of a possible
six on a test, or finding three out of six pearls that slipped off a necklace are not quite the same
thing a s scoring one out of two, or finding one of two missing pearls. In probability, i t i s
calculated t h a t the chances of drawing one ace from a shuffled pack is 4 in 52, but i s t h a t the
same as saying 1 in 13? The context, the use matters. In mathematics, when we say t h a t 316
and 112 are the same, we mean that asgozinto numbers (6 goes-into 3 the same number of times
as 2 goes-into l), they have the same value (see GLOSSARY). In other words, when you convert
them to decimals, and so treat them a s operators on wholes and use them to locate points on a
number-line, they give the same answer.
B
COMPARING I have in mind two fractions with very large numerators and denominators.
Devise a method to enable me to tell which one is larger, or whether they are equal. Try to
minimise the amount of arithmetic t h a t I have to do, with no calculator available or with the
numbers so large that the calculator won't immediately help.
B
Comments It may not be entirely obvious what the problem is. If this is the case, suppose that you meet
someone who knows only how to work with whole numbers, having never heard of fractions. Give them a
rule for testing which of two fractions is the larger, or whether they are equal, using only instructions to do
with integer arithmetic. Eudoxus appears to have been the first to write down a completely satisfactory
answer to this question, in Book X of Euclid.
Whenever there are multiple names for mathematical objects, there i s a potential difficulty in
doing computations. It may be the case t h a t 316 and 214 are names for t h e same number, but
what happens if they are combined in some way with some other number(s). Will the answer
also be the same? If so, all well and good, but if not, the whole idea of arithmetic would break up
into pieces.
FRACTIONATED What number has the property that when two is added to both numerator and
denominator the result is to increase the number by one-tenth, and when four is added to both
numerator and denominator, the result is to increase the number by one-sixth?
b
Comments Note the ambiguity in the language of 'one-tenth'. Does it mean 'one-tenth of the number', or
the number 'one-tenth'? Decide which meaning you wish to consider. Some fiddling around, either
algebraically or with specific fractions, reveals U8 as a solution, but notice that 112 and 214 are not solutions.
The question is a bit of a trick, because although there is afraction that satisfies the condition, there is no such
number. In fact, numbers do not have numerators and denominators - only fractions have them.
INDICES Do xv2 and xY4 always mean the same thing or have the same value? What about xv3
and x3I9?
Comments These questions assume familiarity with indices, and the use of x l n to mean the square root of
r The expression x l n makes sense only when x is not negative. The expression z2I4 could be used with
negative values of X , if z2I4 means ( x 2 ) l i 4 , since the squaring would remove the minus sign. Unfortunately
though, it would be convenient to be able to interchange the orders, so that (x2)li4 = ( x " ~ ) ~which
,
would run
into trouble with X being negative. The expressions x1I3 and f19 make sense for z positive or negative, and
have the same value, as long as you take the real cube root or ninth root as required.
When two fractions such a s 2/4 and 18136 are considered to be different names for t h e same
number, there suddenly arises a nasty question. If you add (or subtract or multiply or divide)
two numbers, does i t matter which fraction-names you use to do the computation? Might you
not get different answers depending on which names you use? The next four activities explore
this question.
RATIONALS Determine a succinct description of all the fraction-names for a given number.
Comments It will be useful for the next few activities to have a clear, concise description of all fractionnames, and for telling when two fraction-names actually name the same number. The word rationals (see
GLOSSARY) is used to describe the numbers which have fraction-names, when fractions such as 214 and
9/18 are taken as being equivalent, in the sense of naming the same number or describing the same
operation.
P
RATIONAL CONSISTENCY Take two equivalent fractions, like 214 and 9/18, and take another
pair such as 6/18 and 361108. Are you sure that if you add 214 and 361108 you will get the same
answer as adding 9/18 and 6/18? Why? Try t o find an argument which will convince a sceptic
and which applies to any choice of fraction-names.
Comments It is not enough to appeal to some claim such as numbers just work that way. One must be
certain that nothing will go wrong. It is also useful to find an argument which does not depend on the
specific examples chosen, but rather which indicates why it always works.
MARKING How could anything go wrong? Try the same activity using the idea of fractions
a s marks, where 2/14 means 2 correct out of a possible 4. Now the adding is different, and all is
not well in adding equivalent fractions! Why not?
P
EXTENDING Repeat the same activity for subtraction, multiplication and division.
P
REFLECTING What are the implications of the last four activities in terms of thinking of
fractions a s numbers? What do your pupils think fractions are (numbers, operators, . . .l?
P
ASSESSING What behaviour on the part of pupils might suggest that they have mastered the use
of fractions a s names for numbers?
P
REAL NUMBERS
A real number is a number which can be located on a number-line as a distance measured
from the origin. Real numbers encompass whole numbers (positive and negative), rational
numbers (each with many equivalent fraction-names), and all the irrational numbers a s well
(everythin left over!). Why does 42 X 43 = 46, and how do you know for sure? Which is
greater: 4252 or 240.5, and how do you know for sure?
These two questions invoke names of numbers which are not decimal-names. Consequently
i t may or may not be appropriate to translate these questions into decimal-name questions. If
you do this, you find yourself comparing or multiplying infinite decimal-names, but it may or
may not be entirely clear just how you go about doing so.
After considerable thought, Dedekind proposed in 1858 that in order to specify a method for
multiplying decimals, it was necessary to go back and re-define what he meant by a decimal
number. His solution was rather sophisticated, but it was based on the idea that to multiply 42
by 43, we take better and better approximations of each, multiply them, and then take the
answers a s being better and better approximations of the true answer. This is the idea behind
trapping (see GLOSSARY).
F
TRAPPING POWERS Decide what 3d2 ought to mean, then generalise to ad2for positive a. Use a
calculator to decide what (3d2)J2
seems to be a s a decimal, and justify your answer using the
laws of indices.
F
Comments Note that this example was chosen so that the answer would be recognisable. The challenge
is to convince yourself that the symbols do indeed represent numbers.
F
MORE TRAPPING Use a calculator to conjecture which of J2d2 and 2d0.5 is the larger, and
convince a colleague that both of them are names of numbers. Use the laws of indices to verify
your conjecture. Is there any other method other than using the laws of indices for telling
whether the two are the same? Compare J 2 + J3 and J 5 in a similar way.
3'
Uncertainty is not confined to arithmetic operations. For example, although i t is fairly easy to
deduce a rule for deciding which of two fractions is the larger, there is some difficulty in
deciding which of two decimal numbers is the larger!
DECIMAL COMPARISON Write down a rule for deciding which of two decimal numbers
(presented a s decimal-names) is the larger. Convince a colleague that your rule is correct.
Comments The usual rule which pupils learn in school is unfortunately deficient in one minor but very
important respect. According to your rule, which of the following decimal-name numbers is the larger:
1.4356 or 1.43559 ? It is not hard to modify the usual rule so that it copes with recurring nines, but any
attempt to outlaw recurring nines altogether fails when there are multiplication or addition sums involved,
because these operations can generate recurring nines as part of the calculation.
I t is typical of working with decimals that you only get to work with an approximate value.
There arises, then, the question of just how approximate is the answer to a computation done
with approximate values.
F
APPROXIMATE ARITHMETIC You are told that a room is so many metres wide and so many
metres long, correct to the nearest centimetre. What is the perimeter of the room, and correct to
the nearest what? What is the area of the room, and correct to the nearest what? Generalise to
addition of several quantities, subtraction and division.
F
Comments Try specific numbers, but then extend to the general case.
The approximations in the last activity were stated in the form 'correct to the nearest. . .'.
This means that, 1000 m correct to the nearest centimetre and 1 m correct to the nearest
centimetre have the same absolute error (see GLOSSARY). Approximations can also be
measured relative to the measurement itself, a s a percentage (see GLOSSARY). Thus, an error
of 2% tells us that a measurement of 1000 m could be anywhere between 1020 m and 980 m, while
a measurement of 1m could be anywhere between 1.02 m and 0.98 m.
F
RELATIVE ARITHMETIC What happens to relative errors (given a s percentage errors) when
numbers are added, subtracted, multiplied or divided? Work out a handy rule for finding the
relative error in the answer, given the relative
error in the starting numbers, that can,be used
. .
when doing such calculations.
. ,
Comments As with the previous activity, some operations behave nicely with respect to relative errors,
and some do not.
ABSOLUTELY RELATIVE What is the absolute error of the arithmetic mean of a collection of
numbers, all with the same absolute error? What is t h e relative error of the geometric mean of
a collection of numbers, all with the same relative error?
F
REFLECTING What are the relative merits of absolute and relative errors, and how is this
reflected in their arithmetic?
F O R THE BOLD Investigate what happens to errors in x raised to th2 power y when
error of p% and y has an error of q%.
x
has an
SURD ARITHMETIC
Arithmetic with surds i s often easier than with fractions. since there is much less
simplification that can take place! For example, remember that the only. thing
..
you know about
43 is that 43 X 43 = 3.
SU~DScompare the following calculations with adding and multiplying a + b and a - b.
Add 5 + 43 and 5 - 43. Check your result on a calculator.
Multiply 45 + 43 by 45 - 43. Formulate a conjecture and generalise.
1
Find another name for
in the form ofp + q43. Formulate a conjecture and
generalise.
m
-
F
M O R E SURDS Multiply 1 + 43 by itself. Deduce the square root of 4 - 243 as well as that
of 4 + 243. Generalise your method to find the square root of other similar expressions.
-43
Using the ideas developed so far, show that
and generalise.
+ ?m
is, in fact, another name for 2,
b
) PRE-ALGEBRA What connections do you see between doing computations with numbers like
43 and 45, and doing computations with 1 + a and 1 + X ? Has your confidence with surds
changed a t all - if so, in what way and why, and if not, why do you think not?
Comments Some people find that it helps their appreciation of algebra to work with 42 etc. a s an
intermediate stage between whole numbers and letters. Others find 42 and z equally abstract. If you see 42
as a name for a number, and x a s a name for a number not yet known by you, there are close similarities.
F
REFLECTING What qualities of surds make them difficult to think about, and what qualities
make them simple to work with?
ABSTRACTION
The establishment of a firm foundation for the arithmetic of decimal number-names, a s the
arithmetic of real numbers, was a long and complicated mathematical journey, involving the
re-thinking of what previously seemed obvious. In this section, a few of the high points and
difficulties were touched upon. It is typical of human beings that they repeatedly extend ideas
(such a s whole numbers to fractions to decimals) and that, when they begin to encounter
difficulties or conflicts in their intuition, they go back to first principles and start asking more
searching questions. A mathematician seeks a precise definition of intuitive terms such a s
'numbers', and how to do arithmetic with them. Decimal-names provide one route to
specifying what is meant by a real number (and a geometrical approach based on the numberline is given in the next section). Trapping provides one way of specifying how to do
arithmetic with such numbers.
REFLECTING What similarities and differences are there in the awkwardnesses of fraction
and decimal arithmetic? Why is this?
What role is played by fractions in your understanding of decimals and their arithmetic?
What similarities and differences are there between arithmetic with whole numbers,
fractions and decimals on the one hand and surds on the other?
b
A S S E S S I N G What behaviour on the part of pupils might suggest that they appreciate the
arithmetic of fractions, of decimals and of their connection? What questions might you use to
probe for this behaviour?
4 DIFFERENT BASES
One of the reasons for stressing the use of the word decimal as a kind of a name for numbers is
that there are many different ways to refer to the same number. The integer 17 can also be
written as 17.00, as 16.9 (see UNIQUENESS OF NAMES on p. 15), as the fraction 10213 and as 12 +
5, and these by no means exhaust all the possibilities. This section is about yet another aspect
of decimal names. Do not try to work through it activity by activity. Rather, find something
attractive, and look at activities near it.
The points on the number-line have all been given decimal number-names, the word decimal
drawing attention to the significant role of ten, but there is no reason to stick with ten. The
idea of different bases is introduced for whole numbers, and then for fractional parts, and then
examples are given of contexts in which the use of different bases is helpful. A more complete
historical study of the emergence of our place-value and the decimal system can be found in
G. Flegg, Numbers: their history and meaning, Penguin 1983.
INTEGERS
Even integers have more than one name and, as this subsection shows, those names can be
useful in particular contexts.
F
THE VALUE OF PLACE Start with a number like 237. What do you see as you think about 237?
You might see it somewhere out on the number-line, you might see it as a whole entity, you
might see the structure of the number as made up of digits. Just what is meant by the 2, the 3 and
the 7 in 237, and why is their role different in 237 from 723?
F
REFLECTING You might have been tempted to respond very quickly to the last activity. Pause
and rehearse all the language patterns that you can think of which might be involved in an
explanation of the base-ten notation for numbers. Consider also the language patterns that
your pupils might use in the same task. Do any of their explanations indicate incomplete or
erroneous ideas?
b
The essence of the base-ten system is that the place-value of a particular position is a power of
ten. The names for the powers two and three, square and cube, indicate and are derived from
the original Greek geometrical way of thinking about number as length, area and volume.
There is nothing special about base-ten, apart from the fact that it is most commonly used in
human communication.
F
CONVERSION Let eight take the place of ten in writing numbers, so that 23T8 means
Rehearse this by converting some base-eight names to base-ten, and then explore the
conversion. of base-ten names to base-eight names. For example, successively dividing the
base-ten name 237 by 8 and recording the remainders gives
m
28
3
0
(remainder 5)
(remainder 5)
(remainder 3)
-
from which the name 3558 can be written down. Convince a colleague why this procedure will
always give the correct answer, and generalise to converting base-ten names to any other
base.
-
Comments The first task is to work out what this procedure actually is, in your own words. To do this, it
helps to try several examples, until you are confident. For example, try converting the base-ten name 159 to
its base-eight equivalent.
F
-
'
- . ..
FOR THE BOLD Work out a method for converting base-two numbers to base-five numbers
directly - that is, without going via base-ten.
b
Comments This activity is an excellent exercise in seeing and then expressing a general pattern or
technique. It is one tEng to 'sort of see how one might go about it' and much more mathematical to get stuck
into some examples in order to become familiar with and confident enough in different bases so that you can
work out and express a technique for conversions.
Base-two and base-eight are used in computers. Second generation machines use base-sixteen
(hexadecimal), which can also be seen on the BBC computer when referring to tracks of the
disc. The digits used in the hexadecimal system are: 0, 1, 2, . . ., 8, 9, A (for 10), B (for l l ) , C
(for 12), D (for 13),E (for 14) and F (for 15).
F
REFLECTING What digits appear in base-eight number-names? Base-seven? Generalise, and
contemplate the requirements of the Babylonian system of base-sixty.
P
Comments The digits are the remainders upon dividing by the base, so the only digits that appear are
digits between 0 and the base. Check your method of conversion by converting the base-sixteen numbername 1AF3,, to its base-ten equivalent.
REFLECTING AGAIN What numbers are represented by 108, 14, 102 and 1016? What naming
system are you using to refer to the numbers? What numbers are represented by the names B8,
135, 10102 and A16?
b
Comments You need some name to use to refer to a number, and the tendency is to use the most familiar
name. Indeed, the tendency is to think of the base-ten name as the number, rather than as a name for a
number. Such identification causes confusion when you encounter the different names for ten. We tend to
think of 10 as ten, when it actually has the meaning 1X (10) + 0 which is subtly different (as a name not a
value) from ten.
SUMS IN BASE-EIGHT Try adding a n d multiplying 1238 a n d 6778 without, of course,
converting to base-ten first! The point i s to reconstruct your method of adding and multiplying
by reinterpreting i t in base-eight, and thereby reminding yourself of what pupils go through
when trying to work i t out the first time.
b
The historical roots of number-names is a fascinating story, and much too long to be recounted
here. (See, for example, G. Flegg, Numbers ancient a n d modern, Macmillan 1989, or
G. Flegg, Numbers: their history and meaning, Penguin 1983.) It is worth remarking that all
sorts of different bases have been used a t different times. The Babylonians used sixty as their
base, presumably because sixty has so many factors that many fractions of a whole have finite
names.
REFLECTING Investigate what exactly is meant by the last sentence, bearing in mind that our
measurement of angles in degrees, minutes and seconds comes from the Babylonians.
b
It is recorded that the great mediaeval mathematician Fibonacci used the extended Babylonian
system to name a root of the equation x3 + 2x2 + 10x = 20, a s 1.22~7ii 42iii 33iv 4V 4ovi where
successive blocks are multiples of powers of one-sixtieth.
The Roman system used twelfths in much the same way that the Babylonian used sixtieths, so
that 3.@7ii meant 3 + 5112 + 71144. They also used words such as uncia (for one-twelfth of an as,
which was a unit of weight), and then subdivided that into twenty-four scrupuli and a
scrupulus into eight calci. Simon Stevius, a Dutch mathematician, appears to have been the
first to use a base of ten rather than twelve or sixty for his calculations in a book published in
1582. He placed a marker (a circle with a number in it) instead of the Roman superscripts used
by Fibonacci.
b
ROMAN CALCULATIONS Work out a method for doing calculations such as:
3 as, 4 uncia, 5 scrupuli, 6 calci
added to 2 as, 9 uncia, 22 scrupuli, 5 calci.
b
Comments Try to specify complete instructions for a Roman pupil. You might also wish to tackle
multiplication, but you will need considerable patience!
REFLECTING What contribution, if any, does working on different bases for whole numbers
make towards strengthening understanding of the base-ten system?
Comments The study of numbers in other bases used to be popular, but has recently gone out of fashion.
You might like to try to explain why this has happened.
PARTS OF A WHOLE
The idea of using different bases can be extended to numbers between 0 and 1. When whole
numbers are being discussed, the word base is used, whereas when numbers which may be
parts of wholes are involved, the word decimal is used, and by extension, the suffix c i m a l .
Otherwise, the extension from decimals to other bases proceeds just as you would expect.
CIMALS Just how would you expect base-two or base-eight to be extended to numbers between 0
and l ? What is the decimal meaning of number-names such as 101.01012 and 3.548?
b
NAMING What is the
l.'
likely to be called in base-two and in base-eight?
b
Comments There is no standard name, but there is also a tendency to (mistakenly) call it a decimal point!
It should presumably be called a bicimal point (two-cimal point) and an eight-cimal point respectively.
Having explored the idea of using different cimals, the question arises a s to how to convert
number-names from one system to another. For example, given the decimal-name of a
number, how do you find the bicimal name? What would a computer program look like to do
this?
CONVERT Explore how to convert number-names from one base to another. A good place to
start would be from bicimal to decimal, and then from decimal to bicimal. Then try tricimal
to bicimal, perhaps by going via decimal a t first. Try to develop a general theory. Since
computers use the hexadecimal system (base-sixteen), you might like to investigate
conversion from sixteen-cimals to decimals and back again!
REFLECTING What is wrong with treating a finite two-cimal like 1.11012 a s a whole number
111012, converting that to base-eight, and then inserting the eight-cimal point?
INFINITE NAMES Rehearse the arguments from Section 1 regarding the name 0.9 in a
discussion of the possible alternative names for 0.i2, 0.i3, and o.'& Convince a colleague.
b
INFINITE vs FINITE NAMES Which numbers (expressed a s fractions or a s decimals if you like)
will have finite bicimal-names and which will have infinite bicimal-names? Generalise to
other bases.
Comments This is another example of an exploration into generality which enables you to see if you
really understand what is going on, and which provides a clear example of an opportunity to see, speak and
then write down a generality.
Mathematicians use the word characterise (see GLOSSARY) in this sort of situation, trying to characterise
those numbers which have finite names in, say, bicimal notation. Obviously one description of such numbers
is 'those numbers with a finite bicimal expansion'. It is desirable, however, to find some way to recognise
such numbers through more familiar properties. Thus, for decimals, fractions whose denominator is
divisible only by powers of 2 and by 5, and by no other prime, have a finite decimal-name.
-
CONVERSIONS FOR THE BOLD Take a decimal number-name, delete all the nines that occur,
and reinterpret the new name as a base-nine name. The result is a new number, so this rule
specifies a function from the reals to the reals. Which numbers stay the same, which get
larger and which get smaller under this mapping?
EXPLOITING DIFFERENT BASES
In this subsection a number of contexts are offered in which the use of different number bases
is useful; bordering on the essential, in order to analyse the situation mathematically.
PRESENT AND NOT PRESENT
The basic notion of twoness, of either being-present or not-being-present, of true or false, of in
or out, of on or off, lies a t the heart of human thinking, and also therefore at the heart of the
functioning of computers. The simplest occurrence is in forming the subsets of a set. Given
the set (A, B, C), the subsets are formed by either choosing or not-choosing each element in
turn. Consequently there are 2 X 2 X 2 = 8 different subsets, and this idea generalises to larger
sets. I t also generalises to computations of, for example, the number of base-ten numbers that
have a t most three digits.
F
HOW MANY POSSIBILITIES? How many base-ten whole number-names are there which have
a t most three digits? How many have a t most four digits? What about in base-three? Careful!
F
Comments Although this activity is not about present and not-present, it is a generalisation. It was
inserted as a reminder that there are many opportunitiesfor mathematical investigation and generalisation,
simply by asking whether something will remain true if you make a change.
In mediaeval Europe there was a standard method for performing multiplications (in the
extreme event in which i t was actually needed) which was still being used in the twentieth
century in parts of central Europe. The method was as follows.
To multiply two numbers such as 27 and 36, successively halve one and double the
other, and keep track of the doubled numbers corresponding to odd numbers in the
halving:
Now add up the numbers in the third column to get the answer 972.
F
PEASANT MULTIPLICATION Check by direct multiplication that this answer is correct, then
reverse the roles of 27 and 36 to see if the same answer arises.
By experimenting with smaller numbers, try to explain why the method works. Make sure you
can state the method in your own words first!
P
Comments Make use of the idea of present and not-present, and powers of two.
F
HALVING Locate the points on the number-line (if any) which meet all of the following
conditions!
bigger than 0
bigger than 1/4
bigger than U4
bigger than 5/16
bigger than 5/16
bigger than 21/64
bigger than 21/64
S
.
.
and
and
and
and
and
and
and
and
smaller than 1/2
smaller than 1/2
smaller than 318
smaller than 318
smaller than 11/32
smaller than 11/32
smaller than 431128
One way to locate the number on the number-line which is trapped by these intervals, is to
observe that a t each stage, the previous interval is divided into two halves, and we are told in
which half the number lies. We could use L to denote the Left half and R to denote the Right
half at each stage. Thus the following numbers in the interval 0 to 1 can be located by
sequences of L s and Rs:
LLL . . . or L is another name for 0 (i.e. always take the left half)
RRR . . .or R is another name for 1 (i.e. always take the right half)
RL and LR are other names for 112
1
P
HALVING (continued) What name in L-R notation should be given to the number in
HALVING? What other more standard names does it have? Investigate the names of other
points which arise from similar procedures.
Comments If you use 0 and 1in place of L and R, and put a point a t the front of the name to indicate that
the number is in the interval 0 to 1,the names you get are recognisable a s bicimals. Notice that the presence
of a 0 or a 1in a given piLe in a bicimal tells you something about the location of the corresponding numberline point, vis-a-vis the halving of a particular interval. This is analogous to the information provided by the
digit in a particular place in a decimal; it tells you where the corresponding number-line point lies in a certain
small interval. It is worth pondering this slightly different view of what a decimal-name means, and trying
to express i t to a colleague (or unsuspecting friend!).
REFLECTING What extension of Ls and Rs would extend the bicimal idea of left- and righthalf to decimal notation? How could the language of 'present' and 'not-present' be used to
describe the process lying behind HALVING?
P
NIM The game of N I M was popular in the 19th century, and was re-popularised in the sixties
in the film 'Last Year At Marienbad'. Players confront one or more piles of matches. In this
case there are three piles.
1111
Ill
11
Players take it in turn to remove any number of matches they wish from any one pile. The
player who is unable to move (because all the matches are gone) loses.
Play the game several times, preferably with a colleague, using different starting piles in
order to get a sense of what happens.
Comments It is well worth specialising: to one pile with arbitrarily many matches; to lots of piles each
with one match; to two piles with the same number of matches and so on. Be systematic, and then every so
often branch out and try something quite different!
NIM (continued) Think of the last move. What positions force a loss immediately? Now
work backwards, building up positions which force the player who confronts them to lose
(assuming the other player plays well and takes every opportunity to win). Try to see some sort
of pattern in the losing positions and in the best reply to the loser's move.
Detecting a pattern is not a t all easy. Observe that if you can mimic or mirror the other
player's moves, then you win, because if your opponent can move, so can you. Thus, having just two piles
exactly the same is a losing position. So is having one pair of identical piles and another pair of identical piles
(but not necessarily the same a s the first two). More generally, a pile can be thought of a s consisting of subpiles, and if your opponent takes some from several sub-piles of the same pile you could take a matching
amount from the same sub-piles. This idea can be developed into a strategy which removes the game
element, since one player can always win.
Comments
The act of weighing makes subtle use of present and not-present (which is a base-two idea),
and also of the base-three idea.
P
WEIGHTS Suppose you are asked to design a weighing system, in which you put known
weights on one pan, and a weight to be determined on the other, and so that you can weigh to the
nearest uncia (see p. 41) anything from 1uncia to 4 a s (there are 12 uncia in an as). How few
weights can you provide so that this can be done? What is the connection, if any, with present
and not-present?
Comments Base-two is of great assistance in designing such a set of weights. Why?
DEFECTIVE (one) Imagine that you are presented with a number of identical looking objects,
and told that they are all supposed to weigh the same amount, namely W ,but that one object is
defective and weighs a different amount. Your task is to determine the defective object with as
few weighings a s possible. How many weighings are needed for n objects?
P
Comments Specialise to a few objects before generalising to many! Think perhaps about the theme of
present and not-present.
DEFECTIVE (continued) You can cut down the size of the set of objects which contains the
defective one, by one-half a t each stage: weigh half of the set and decide which half the object
lies in. Indeed, if the objects are numbered in advance, and if a t each stage when you choose
half to weigh, you choose the smallest numbered objects still undetermined, then a t each stage
if you record with a I or an 0 whether the defective object is 'in' or 'out' of the weighed group, the
sequence of I s and OS can be decoded to tell you the number of the defective object. How?
Comments Specialise to small numbers and then generalise. Bear in mind present and not-present.
The Towers of Hanoi puzzle (also known as the Towers of Brahma) purports to come from a
monastery in Hanoi, in which there are three golden needles and a stack of 100 discs, each of a
different size, and each with a hole in the middle so that it fits on the needles. At the beginning
of the universe, the discs were all neatly piled on one needle, from the largest a t the bottom, to
the smallest a t the top. Each day the monks transferred one disc to a different needle, subject
only to the constraint that they must never place a disc on top of a smaller disc. The question
then is 'how many moves will it take to get all the discs piled up on one of the other needles (not
the starting one)?'
In fact, the puzzle seems to have been invented by the French game-constructor Lucas in the
19th century, but i t has proved so popular that you can still buy versions. Psychologists have
studied in great detail how people solve the problem of transferring the discs (only 4 or 5 of
them!) from one needle to another. The main problem is keeping track of what you are doing
and where you are going.
TOWERS OF HANOI Try out the puzzle with a few discs in order to explore potential
difficulties. You can make a simple version by using stacks of different sized coins, or pieces
of paper, and dispense with the needles! One way to keep track of your position is to represent
each disc by a different power of two. Then you can follow your moves by following the binary
representation of the discs on each needle.
If you are familiar with this puzzle, try i t with three stacks of discs, each stack having its own
colour. Decide which 'needle' is to receive which colour stack, and then try to achieve this
using the rule that a disc is never placed on top of a smaller disc. Alternatively, number N
pieces of paper with the numbers 1to W, shuffle them, and make three random piles; then sort
them into order using the rule that no number may be placed on top of a smaller number.
Comments You are, of course, more interested in whether it can always be done, and if so, how, rather
than whether it can be done in your particular case.
) REFLECTING What connections are there between the theme of present and not-present and
TOWERS OF HANOI?
TRlClMALS AND BASE-THREE
Although we are very familiar with the binary situation of yes and no, present and not-present,
odd and even, the English language does not offer similar words for situations which come in
threes.
Bicimals a r e the analogue of decimals, but using only zeros and ones, a n d a r e based on
halving. A reasonable conjecture will probably have formed in your mind a s to how tricimals
relate to trisecting intervals. Use the HALVING activities a s a model to design and explore
tricimals and trisections.
MORE WEIGHING Again you have been asked to design a weighing system to measure
weights of from 1 uncia to 3 as (12 uncia make one as) in steps of 1 uncia, but now you are
permitted to put weights on both pans if you wish. How few weights can you use, and what is the
connection with tricimals?
b
Comments Convince a colleague that you see why base-three enters this situation. Generalise to
measuring up to larger numbers of as.
DEFECTIVE (two) You are confronted with a pile of objects, all looking the same and supposed
to weigh the same amount, but are told that one of them is defective. You only have a balance.
How can you locate the defective object most efficiently?
Comments Make use of the ideas from the previous weighing activities, and make use also of the fact that
there are two pans to a balance, so that one lot can go on each pan, and one lot can be put aside. Specialise in
order to generalise to any number of objects. Weighing problems are common in mathematical recreation
books. Often you are asked to put the objects in order of increasing weight, using a balance and in as few
weighings as possible. That is the hardest and still not completely solved version. In other versions you
might be told there are two defective objects, or three . . .. There is no guarantee that the tricimal idea will
help when the problem changes!
The rational numbers are scattered all over the number-line, to the extent t h a t you can get a s
close a s you want to a n y number and still be standing on a rational point. This i s what lies
behind the use of fractions a s approximations which trap numbers which need infinite but nonrepeating decimal-names. This idea is an important one in mathematics: one direction for
exploration and extension is explored in the following activity.
DENSE Let C be the set of all tricimals between 0 and 1 which have no 1in their tricimal name.
Let D be the set of all other numbers between 0 and 1. Form a (rough) picture of where on the
unit interval the points D lie, and hence form a (very rough) picture of where the points in C lie.
Observe that the tricimals
0.012,
0.0122,
0.01222,
0.012222,
...
all belong to D , but that they get closer and closer to 0.02 which is in C. Indeed, you can get as
close as required to 0.02 by going far enough down the sequence. Pick a tricimal name of a
number which lies in C, see if there is a sequence of numbers in D which gets closer and closer
to it, indeed a s close a s anyone wishes no matter how close that might be. Can you find any
numbers in C for which there is no sequence of members of D which get a s close a s you wish to
it?
Comments Just as any number can be approximated by a sequence of terminating decimals (rational
numbers), so any number can also be approximated by a D number. Sets such as D or the rationals having
this property are said to be dense (see GLOSSARY) because no matter where you look on the number-line,
there must be some of these numbers nearby.
ABSTRACTION
One of the reasons for stressing decimals as names for numbers rather than as the numbers
themselves is that the notation used to name a number is sometimes helpful, and sometimes
not. Thus when thinking about dividing intervals into halves, over and over again, the
bicimal notation is much more useful than the decimal. Furthermore, by stressing the name
aspect, it is easier to distinguish properties of numbers from properties of names. Thus, the
property of having a finite decimal-name (or of ending in recurring zeros) is not so much a
property of the number a s of the number-name. The property of being divisible by three is more
a property of the number, and it may be more or less visible in the particular naming system
used.
In recent government statements about mathematics teaching, attention has been diverted
away from using different bases, whereas previously (and especially with pounds, shillings
and pence), it was thought useful to be able to work with different bases. The observation that
familiarity with different bases could help highlight the role of place-value in the decimal
system, and hence support deeper understanding of decimals has not always been borne out in
practice. When the use of different bases becomes something to be taught, rather than explored,
and facility in computation is stressed over comprehension, pupils naturally struggle to
master techniques and are less likely to encounter the awareness-shifting aspects intended.
) REFLECTING What are the pros and cons of getting pupils to explore different bases? What
might they get from such work, and how might that be encountered in some other way?
ASSESSING What behaviour on the part of pupils might suggest that they appreciate the role of
ten in the decimal system of naming numbers? What activities might you use to probe for this
behaviour?
b
5 GEOMETRICAL ARITHMETIC
The purpose of this section is to demonstrate that arithmetic operations on numbers can all be
undertaken geometrically. This is no mere idle exercise. It is an attempt to decide just what a
number really is. It is thought that the Greek sense of number was dominated by the image of a
line-segment, so that their arithmetic thinking was rooted in geometry. Certainly, they
answered the question 'what is a number? with a geometrical definition, a s a position on the
number-line relative to a fixed origin. I t was necessary therefore to be able to show that it is
possible to calculate using geometrical constructions. Today, especially with electronic
calculating devices, we are dominated by the arithmetic of number, and struggle to retain a
sense of the geometric.
Thinking of numbers a s line-segments (which amounts to thinking of numbers purely a s
points on a number-line) makes addition perfectly easy, and subtraction (Euclid, Book I,
proposition 111) is much the same:
Before going on to multiplication, note that i t is easy to divide a given line-segment into a
whole number of equal pieces, by making use of 'repeated addition' and parallel lines:
) REFLECTING Explain to yourself, or to a colleague, how you would go about dividing a linesegment into a given number of pieces.
b
Multiplication is, however, rather tricky. I t is believed that the Greeks thought of numbers as
having dimensions, so that a product was seen a s an area, usually the area of a rectangle. The
product of three numbers would then be the volume of a cuboid. This idea is still present in the
names that we use for squaring and cubing, and explains or a t least indicates why Dienes
apparatus (Multibase blocks) is used to help pupils appreciate the place-value aspect of our
number notation. One weakness is that i t becomes hard to think about four or more numbers
multiplied together.
To multiply two line-segments, form a rectangle as follows:
The result of such a product is an area, not a length, so some way is needed to convert an area to
a length.
CONSTRUCTION 1 To find a rectangle of the same area as a given rectangle, and with a given
line-segment a s one edge, go through the steps suggested by the following diagrams:
How does this construction help you to multiply two line-segments and get a line-segment -as a
result?
DIVISION How can you divide one line-segment ,by another?
,
P
l
CONSTRUCTION 2 Areas are liable to arise from triangles and more general polygons as the
results of various constructions. To deal with these in segment arithmetic, it is necessary to be
able to convert areas of polygons to something more manageable. To convert a triangular
area to a rectangular area, go through the steps suggested by the following diagrams:
How might this construction together with CONSTRUCTION 1 be used to convert a triangular
area to a line-segment? How might you then convert the area.of a polygon to a line-segment?.
i
Comments There are at least two ways of going about this, depending on how eager you are to add up
segments rather than rectangles.
GENERALISING Generalise CONSTRUCTION 2 to show that the area of any triangle can be
presented a s the area of a parallelogram with a given interior angle (Euclid, Book-I,.
proposition XLII). Generalise further to show that one edge-length can also be selected in
advance (Euclid, Book I, proposition XLIV).
P
SIMILAR PRODUCTS Another way to find the product of two segments as a segment involves
the use of parallel lines. Use the following diagrams to work out such a method.
Comments There is some evidence to suggest that Euclid and his contemporaries saw this construction
as quite different in nature to the area construction. What differencesdo you detect?
SOLVING EQUATIONS Outline constructions for solving the general equation ax = b where a
and b are given a s lengths of line-segments. What about negative values for a andor b?
P
SQUARE ROOT CONSTRUCTION In order to solve quadratic equations, i t is certainly
necessary to be able to find square roots. Use the diagrams overleaf to tell a colleague how to
find a segment whose square is the product of two given segments.
P
This construction (Euclid, Book IV, proposition XIV) led to the famous Greek questions of
whether there is a similar construction for cube roots, and whether i t is possible to construct a
square with the same area a s a given circle (otherwise known as squaring the circle). I t was
not until the 19th century that mathematicians were able to prove that neither of these
constructions are possible.
Finding a square equal in area to a rectangle is (in our algebraic way of thinking)
tantamount to solving a simple quadratic equation. To solve other quadratic equations, the
Greeks needed to devise special techniques. For example, the most famous Greek theorem,
Pythagoras' theorem, shows that the sum of two squares can itself be represented as a square
(Euclid, Book I, proposition XLVII), a s follows:
Pythagoras' theorem can then be used to solve other particular quadratics, such as in the
following construction.
SOLUTION To divide a line-segment into two parts so that the rectangle of the whole with one
part is the same a s the square on the other part (Euclid, Book 11, proposition XI), go through the
steps suggested by the following diagrams:
State what you believe to be the method of solution here (you may wish to modify your conjecture
more than once). It may help to interpret the problem algebraically, and then to use that to
50
interpret the construction in order to verify that it does provide a solution. The following
diagrams indicate how Euclid went about a proof that the construction does what is asked.
Comments This particular problem has considerable significance, since the ratio of the solution segment
added to the original segment is the famous golden ratio which appears a s the chord of the regular
pentagon, the proportions of the Parthenon and the limiting ratio of successive terms of the Fibonacci
sequence, to name but a few. You might like to try solving other quadratic equations in the Greek style.
REFLECTING What differences in thinking about numbers arise from trying to take the Greek
perspective of number as length of a line-segment? What insights if any does it offer that are
otherwise lost or overlooked in our calculator age?
6 THE LAST WORD
Work on this pack cannot be considered to be completed unless and until some overall sense
has been made. The title, DEALING WITH DECIMALS, suggests that mathematicians have found
themselves having to re-think what exactly is meant by a decimal as a result of running up
against such awkwardnesses a s infinite decimal-names, several different decimal-names
for 'the same number, and uncertainty a s to how to do arithmetic with infinite decimal-names.
P
I
LOOKING BACK What is your sense of decimals? How do decimals relate to whole numbers,
integers and fractions? Compare notes with colleagues. Re-read the INTRODUCTION, and see
how you respond now to the questions posed there.
What shifts or modifications in your sense of decimals have taken place while working on
this pack? What are the implications for your classroom?
SAME AND DIFFERENT Think back over the various activities worked on in this pack. What
is similar about them, and what is different? How do they relate to awareness, imagery and
misconceptions; to language patterns and techniques; to contexts and motivation (see the
bookmark from any of the PREPARING TO TEACH packs in the MATHEMATICS UPDATE series)?
CONSTRUING AND MAKING SENSE Make a list of some of the technical terms to do with
decimals you can recall using or coming across. Now try to construct sentences which
connect these terms together, and which describe what you found happening in the various
activities you encountered.
P
Comments This technique of reconstruction of a topic, starting from technical terms, is useful at the end
of a lesson, at the end-ofa sequence of lessons, and before an examination. It is intended to remind pupils
that they are the ones who must make sense of the ideas, and that trying out their story on colleagues gives
them a chance to modify and develop their story, and to make it more fluent.
AIMS Look back a t the list of aims for this pack (see p. 5). To what extent has each aim been
achieved? What are the gaps, and how might something be done about them? To what extent
have your own aims been met?
IN THE CLASSROOM Choose any topic you teach, and look for aspects of decimals which are
present in or assumed in that topic. What sorts of difficulties are likely to emerge through
unfamiliarity with, for example, multiple decimal-names for numbers?
What activities might be particularly effective in helping pupils demonstrate that they have a
good sense of decimals?
b
7 GLOSSARY
ABSOLUTE ERROR A measurement is usually accompanied by an indication of the accuracy
of the measurement. For calculations involving
- addition and subtraction, it is usual, and
easier, to express the error a s an absolute number, a fixed quantity such a s 3.4 0.05,
indicating the interval within which the true value has been trapped.
+
BASE The word base is used to refer to the place-value system being used to name numbers.
We normally use a base of ten, but other bases are also possible.
BISECTION METHOD The method of trapping a number and determining a s many decimal
digits a s required by successively halving the width of an interval in which the number is
trapped is called the bisection method.
CHARACTERISE The process of characterising involves looking for properties of some objects
and then proving that any object satisfying those properties must be one of the objects you had in
mind.
ClMAL Cimal is a generic term used to speak about a way of naming numbers using bases
other than ten, and extending this to expressing parts of a whole in bases other than ten. Thus,
decimal becomes bicimal (base-two), tricimal (base-three) etc., and 'decimal point' becomes
'bicimal point' etc.
.
CLOSER AND CLOSER TO. The expression closer and closer to is an informal way of
describing the mathematical idea of a limit. A sequence of numbers gets closer and closer to a
given number if, no matter how close you require, the sequence eventually gets a t least that
close, and stays a t least that close to the given number.
CORRECT TO SO many DECIMAL PLACES It is usual when using numbers to specify physical
measurements, to use only a finite number of decimal places. Thus, the measurement 3.1234
is correct to four decimal .places if the true measurement is a t least a s big as 3.12335 and
smaller than 3.12345. The number 3.0 is taken to signify being correct to one decimal place;
3.0000 is correct to four decimal places. See also rounding and truncating.
COUNTING NUMBER The numbers denoted by 1, 2, 3,
counting numbers or the natural numbers.
. . ., used for counting, are called the
DENSE A set S is said to be dense in the interval [0, l ] if every real number in the interval is
approached a s closely a s desired by a sequence formed by members of S. In other words, for
every real number r, there is a sequence formed from members of S which converges to r.
GOZINTO NUMBERS Three goes-into six twice, so 'twice' is a gozinto number.
INTEGER A number, named by symbols such as 3, 7, 0, -2, 139, -998 etc., is called an integer.
Integers are t h e result of extending the idea of counting or natural numbers to include zero and
the negatives.
IRRATIONAL Numbers (such a s 45, or x ) which are not rational are said to be irrational.
NUMBER Originally the word number meant a counting 'number, but a s people have become
.-
more sophisticated in the questions that they wish to answer, it has been necessary to extend the
idea of number to include zero,-negatives, fractions, decimals, and more.
OPERATION An arithmetic calculation signalled by +, -, X, + or / is called an operation in the
sense that i t signals an operation to be done with two numbers, giving a single number as the
answer. The term operation is usually used when there are two input numbers. (Technically,
this is a binary operation.)
OPERATOR An action performed on numbers such a s 'adding 3', or 'multiplying by 0.7', or
'taking two-thirds-of is called an operator since it operates on numbers to produce another
number. The term operator is usually used when there is only one input number.
(Technically, this is a unary operation.)
PERIOD The period of a recurring or periodic decimal-name is the length of the repeating
sequence of digits.
PERIODIC A decimal (or other cimal name) is periodic or recurring if the tail end of the
number is formed by repeating the same (finite) sequence of digits over and over again. 0.3
illustrates the simplest form and 2.846b3857'6 illustrates the most general form. Any number
with a periodic decimal-name must be rational.
RATIONAL A number which arises as the ratio of two whole numbers is said to be rational. It
signifies the value common to many different fractions.
REAL NUMBER Real numbers constitute all numbers which can be located on a number-line,
and include all the whole numbers (positive and negative), the rationals, and the irrationals.
RECURRING See periodic.
RELATIVE ERROR A measurement is usually accompanied by an indication of the accuracy of
the measurement. For calculations involving multiplication and division, it is usual, and
easier, to express the error as a percentage of the measurement itself. Such an error is called a
relative error.
ROUNDING A decimal number can be rounded to a given number of decimal places as
indicated by the following example. To round a decimal number to two decimal places: look at
the digit in the third decimal place; if this is less than 5, truncate the number to two decimal
places; if the digit in the third decimal place is greater than or equal to 5, add one to the digit in
the second decimal place, then truncate the number to two decimal places. Thus 3.246 rounds
up to 3.25, whereas 3.2449 rounds down to 3.24.
SURD A number-name, expressed in the form of a whole number or fraction times a root
(square-root, cube-root etc.) is called a surd. Most surds are irrational. The exceptions are 44,
49 etc.
TRAPPING A number can be trapped inside an interval, and the trapping interval can be made
smaller and smaller, indeed as small as we please. This makes it possible to locate or name a
number approximately but as accurately as necessary.
TRUNCATING A decimal number can be truncated to a given number of decimal places
simply by removing all subsequent decimal digits.
WHOLE NUMBER Often taken to be the same as integer, the term whole number can also mean
a counting number. The word 'number' often conjures up the idea of a whole number, whereas
mathematicians usually use 'number' to mean real number.
STUCK?
Good! RELAX and ENJOY it!
Now somethina can be learned
Sort out
What you KNOW
and
What you WANT
SPECIALISE
GENERALISE
Make a CONJECTURE
Find someone to whom to exp
why you are STUCK
WHAT TO DO
WHEN YOU ARE STUCK!
The following suggestions do not constitute an algorithm. They have been found helpful by others,
but they will only help you if they become meaningful. The way to learn about being stuck is to
notice not only what helped to get you going again, but also what contributed to you getting stuck in
the first place. Such 'learning from experience' is then available for use in future situations.
F
Recognise and acknowledge that you are STUCK
Record this in your working as STUCK! In so doing, you will have broken out of the familiar
experience of going round in circles, retreading unfruitful ground, and will have focussed your
energy and attention on devising a strategy to get unstuck.
F
Write down the headings KNOW and WANT
Under KNOW, make a list of everything that you know that is relevant. Where helpful, replace
technical terms with your own words and include some examples.
Under WANT, write down your current question in your own words. You may need to go back to the
last time you wrote down a CONJECTURE to see if you have lost sight of where you are in the
question. Hence the value of recording conjectures as you work.
Your new task is to construct a bridge, an argument linking KNOW and WANT. Sometimes the act of
listing under these headings will be sufficient to free you from what it was that was blocking
progress. Sometimes you will need to narrow down your question to a sub-question that you feel you
can tackle, or you may find it useful to articulate the prompts 'If only I can show/get/do . . .'. If you
are unable to progress, you may need to SPECIALISE further.
F
SPECIALISE
Replace generalities in KNOW and WANT with particular examples or cases with which you are
confident. Try to get a better picture of what is going on in the particular cases, with an eye to
generalising later.
SPECIALISING has two functions:
to enable you to detect an underlying pattern which can lead to a generalisation, perhaps in the
form of a CONJECTURE;
to simplify a question which is giving you trouble to a form in which progress can be made,
leading to a fresh insight on your original problem.
Be SYSTEMATIC, and collect the data or examples together efficiently. A pattern is less likely to
emerge from random specialising, or from a jumble of facts and figures. Draw clear diagrams where
appropriate.
SPECIALISE DRASTICALLY by simplifying wherever possible in order to find a level at which progress
can be made. Sometimes this involves temporarily relaxing some of the conditions in the question.
If you are still stuck, you may need to TAKE A BREAK
Simply freeing your attention from the problem - or explaining it to someone else - can lead to a
falling away of the block.
Copyright O 1989 The Open University
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